Computational Fluid Dynamics 2010

Computational Fluid Dynamics 2010 Alexander Kuzmin Editor Computational Fluid Dynamics 2010 Proceedings of the Sixth...

Author: Alexander Kuzmin

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Computational Fluid Dynamics 2010

Alexander Kuzmin Editor

Computational Fluid Dynamics 2010 Proceedings of the Sixth International Conference on Computational Fluid Dynamics, ICCFD6, St Petersburg, Russia, on July 12-16, 2010

With 516 Figures

123

Editor Alexander Kuzmin St Petersburg State University Laboratory of Aerodynamics St Petersburg Russia [email protected]

ISBN 978-3-642-17883-2 e-ISBN 978-3-642-17884-9 DOI 10.1007/978-3-642-17884-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011923661 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L., Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This book contains proceedings of the Sixth International Conference on Computational Fluid Dynamics (ICCFD6). The ICCFD conference series was formed in 2000 following the merger of the International Conference on Numerical Methods in Fluid Dynamics founded by K. Oshima in 1985 and International Symposium on Computational Fluid Dynamics founded by N.N. Yanenko and M. Holt in 1969. The conferences have been recognized as the leading international events for scientists, mathematicians and engineers specialized in the numerical simulation of fluid and gas flows. The Sixth Conference, which was held in St. Petersburg, Russia, from July 12 through 16, 2010, provided a forum for the dissemination of technical information in all areas of up-to-date CFD, from basic algorithms to applications in various industries. The highlights of the program were five Invited Lectures and four Keynote Lectures given by renowned CFD experts, who presented a great deal of valuable information and exciting topics for discussions. In the beginning of 2010, the Organizing Committee received over 280 abstracts of regular papers for consideration. Four groups of reviewers, which were headed by D. Kwak (USA), C.-H. Bruneau (France), N. Satofuka (Japan), and A. Kuzmin (Russia), performed a careful peer-review process. The final conference program included 126 oral and 30 poster presentations. The total number of ICCFD6 attendees was 171 from 29 countries. It is worth mentioning that 40 presentations were delivered by Ph.D. students. I would like to express a deep gratitude to the National Aeronautics and Space Administration and the European Office for Aerospace Research & Development for their valuable sponsorships offered to support the conference and publication of these proceedings. I also would like to thank the staff of the St. Petersburg Society for Science & Technology for their contribution to the organizing work. Special thanks are due to graduate students of St. Petersburg State University and undergraduate students of Baltic State Technical University for their efforts to ensure the smooth running of the conference. St. Petersburg, Russia November 2010

Alexander Kuzmin

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Contents

Part I Plenary Lectures The Expanding Role of Applications in the Development and Validation of CFD at NASA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David M. Schuster

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Thermodynamically Consistent Systems of Hyperbolic Equations . . . . . . . 31 S.K. Godunov Part II Keynote Lectures A Brief History of Shock-Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Manuel D. Salas Understanding Aerodynamics Using Computers . . . . . . . . . . . . . . . . . . . . . . . 55 Mohamed M. Hafez Part III High-Order Methods A Unifying Discontinuous CPR Formulation for the Navier–Stokes Equations on Mixed Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Z.J. Wang, Haiyang Gao, and Takanori Haga Assessment of the Spectral Volume Method on Inviscid and Viscous Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Oussama Chikhaoui, Jérémie Gressier, and Gilles Grondin Runge–Kutta Discontinuous Galerkin Method for Multi–phase Compressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Vincent Perrier and Erwin Franquet vii

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Energy Stable WENO Schemes of Arbitrary Order . . . . . . . . . . . . . . . . . . . . 81 Nail K. Yamaleev and M.H. Carpenter Part IV Two-Phase Flow A Hybrid Method for Two-Phase Flow Simulations . . . . . . . . . . . . . . . . . . . . 91 Kateryna Dorogan, Jean-Marc Hérard, and Jean-Pierre Minier HLLC-Type Riemann Solver for the Baer–Nunziato Equations of Compressible Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Svetlana A. Tokareva and Eleuterio F. Toro Parallel Direct Simulation Monte Carlo of Two-Phase Gas-Droplet Laser Plume Expansion from the Bottom of a Cylindrical Cavity into an Ambient Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Alexey N. Volkov and Gerard M. O’Connor Study of Dispersed Phase Model for the Simulation of a Bubble Column . 113 F. Sporleder, Carlos A. Dorao, and H.A. Jakobsen Application of Invariant Turbulence Modeling of the Density Gradient Correlation in the Phase Change Model for Steam Generators . . . . . . . . . . 121 Yasuo Obikane and Shigeru Ikeo Part V Algorithms Reformulated Osher-Type Riemann Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Eleuterio F. Toro and Michael Dumbser Entropy Traces in Lagrangian and Eulerian Calculations . . . . . . . . . . . . . . 137 Philip L. Roe and Daniel W. Zaide Automatic Time Step Determination for Enhancing Robustness of Implicit Computational Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Chenzhou Lian, Guoping Xia, and Charles L. Merkle Part VI Boeing–Russia Cooperation Fifteen Years of Boeing–Russia Collaboration in CFD and Turbulence Modeling/Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Sergey V. Kravchenko, Philippe R. Spalart, and Mikhail Kh. Strelets

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LES-Based Numerical System for Noise Prediction in Complex Jets . . . . . 163 Mikhail L. Shur, Andrey V. Garbaruk, Sergey V. Kravchenko, Philippe R. Spalart, and Mikhail Kh. Strelets

Part VII Discontinuous Galerkin Methods Local Time-Stepping for Explicit Discontinuous Galerkin Schemes . . . . . . 171 Gregor Gassner, Michael Dumbser, Florian Hindenlang, and Claus-Dieter Munz Multi-dimensional Limiting Process for Discontinuous Galerkin Methods on Unstructured Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Jin Seok Park and Chongam Kim Runge Kutta Discontinuous Galerkin to Solve Reactive Flows . . . . . . . . . . . 185 Germain Billet and Juliette Ryan Comparison of the High-Order Compact Difference and Discontinuous Galerkin Methods in Computations of the Incompressible Flow . . . . . . . . . 191 Artur Tyliszczak, Maciej Marek, and Andrzej Boguslawski On the Boundary Treatment for the Compressible Navier–Stokes Equations Using High-Order Discontinuous Galerkin Methods . . . . . . . . . 197 Andreas Richter and Jörg Stiller Analytical and Numerical Investigation of the Influence of Artificial Viscosity in Discontinuous Galerkin Methods on an Adjoint-Based Error Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 Jochen Schütz, Georg May, and Sebastian Noelle

Part VIII Vortex Dynamics Analysis of a Swept Wind Turbine Blade Using a Hybrid Navier–Stokes/Vortex-Panel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Kensuke Suzuki, Sven Schmitz, and Jean-Jacques Chattot Using Feature Detection and Richardson Extrapolation to Guide Adaptive Mesh Refinement for Vortex-Dominated Flows . . . . . . . . . . . . . . . 219 Sean J. Kamkar, Antony Jameson, Andrew M. Wissink, and Venkateswaran Sankaran

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Triple Decomposition Method for Vortex Identification in TwoDimensional and Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 V. Koláˇr, P. Moses, and J. Šístek Part IX Design Optimization Multi-point Optimization of Wind Turbine Blades Using Helicoidal Vortex Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Marcel Wijnen and Jean-Jacques Chattot Performance Evaluation of the Numerical Flux Jacobians in Flow Solution and Aerodynamic Design Optimization . . . . . . . . . . . . . . . . . . . . . . . 241 Alper Ezertas and Sinan Eyi Design Optimization in Non-equilibrium Reacting Flows . . . . . . . . . . . . . . . 247 Sinan Eyi, Alper Ezertas, and Mine Yumusak Shock Control Bump Design Optimization on Natural Laminar Aerofoil . 253 D.S. Lee, J. Periaux, K. Srinivas, L.F. Gonzalez, N. Qin, and E. Onate Uncertainty Analysis Utilizing Gradient and Hessian Information . . . . . . . 261 Markus P. Rumpfkeil, Wataru Yamazaki, and Dimitri J. Mavriplis Part X Interface Tracking Augmented Lagrangian/Well-Balanced Finite Volume Method for Compressible Viscoplastic Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 Didier Bresch, Enrique D. Fernandez-Nieto, Ioan Ionescu, and Paul Vigneaux Stable Simulation of Shallow-Water Waves by Block Advection . . . . . . . . . 279 Lai-Wai Tan and Vincent H. Chu Flood-Waves Simulation by Classical Method of Consistent Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Tao Wang, Lai-Wai Tan, and Vincent H. Chu Part XI Vehicles and Vortices Numerical Simulation and Control of the 3D Flow Around Ahmed Body . 297 C.-H. Bruneau, E. Creusé, D. Depeyras, P. Gilliéron, and I. Mortazavi

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Aerodynamic Multiobjective Design Exploration of Flapping Wing Using a Navier–Stokes Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Yuuki Yamazaki, Akira Oyama, Taku Nonomura, Kozo Fujii, and Makoto Yamamoto Study into Effects of Vortex Generators on a Supercritical Wing . . . . . . . . 309 Jingbo Huang, Zhixiang Xiao, and Song Fu Part XII Turbulence Modeling Numerical Simulation of the Turbulent Detached Flow Around a Thick Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 C. Tenaud, Y. Fraigneau, and V. Daru Progress in DES for Wall-Modelled LES of Complex Internal Flows . . . . . 325 Charles Mockett and Frank Thiele Enhancement of the Performance of the Partial-Averaged Navier – Stokes Method by Using Scale-Adaptive Mesh Generation . . . . . . . . . . . . . . 333 B. Basara and Z. Pavlovic Sensitizing Second-Moment Closure Model to Turbulent Flow Unsteadiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Robert Maduta and Suad Jakirlić Part XIII Turbulent Compressible/Hypersonic Flows High Order Scheme for Compressible Turbulent Flows . . . . . . . . . . . . . . . . . 351 Christelle Wervaecke, Héloïse Beaugendre, and Boniface Nkonga Receptivity of Hypersonic Flow over Blunt-Noses to Freestream Disturbances Using Spectral Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 Kazem Hejranfar, Mehdi Najafi, and Vahid Esfahanian Part XIV Bio-Fluid Mechanics Flow Structures in Physiological Conduits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 A.M. Gambaruto and A. Sequeira Fluid Mechanics in Aortic Prostheses After a Bentall Procedure . . . . . . . . . 371 M.D. de Tullio, L. Afferrante, M. Napolitano, G. Pascazio, and R. Verzicco

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Optimisation of Stents for Cerebral Aneurysm . . . . . . . . . . . . . . . . . . . . . . . . . 377 C.J. Lee, S. Townsend, and K. Srinivas Micron Particle Deposition in the Nasal Cavity Using the v2 -f Model . . . . 383 Kiao Inthavong, Jiyuan Tu, and Christian Heschl Part XV Meshing Technology Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Marshall R. Gusman, Jeffrey A. Housman, and Cetin C. Kiris Mesh Quality Effects on the Accuracy of Euler and Navier–Stokes Solutions on Unstructured Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Aaron Katz and Venkateswaran Sankaran Analysis of a RK/Implicit Smoother for Multigrid . . . . . . . . . . . . . . . . . . . . . . 409 R.C. Swanson, E. Turkel, and S. Yaniv Anisotropic Adaptive Technique for Simulations of Steady Compressible Flows on Unstructured Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 O. Feodoritova, D. Kamenetskii, S.V. Kravchenko, A. Martynov, S. Medvedev, and V. Zhukov Part XVI Aeroacoustics Euler – Navier–Stokes Coupling for Aeroacoustics Problems . . . . . . . . . . . . 427 Michel Borrel, Laurence Halpern, and Juliette Ryan Computational Aeroacoustics on a Small Flute Using a Direct Simulation . 435 Yasuo Obikane An Iterative Procedure For the Computation of Acoustic Fields Given by Retarded-Potential Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Florent Margnat Innovative Tool for Realistic Cavity Flow Analysis: Global Stability . . . . . . 449 Fabien Mery and Gregoire Casalis

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Part XVII Large-Scale Computations Comparison of Parallel Preconditioners for a Newton-Krylov Flow Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Jason E. Hicken, Michal Osusky, and David W. Zingg Large-Scale CFD Data Compression for Building-Cube Method Using Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Ryotaro Sakai, Daisuke Sasaki, and Kazuhiro Nakahashi Part XVIII Euler Solvers A Conservative Coupling Method for Fluid-Structure Interaction in the Compressible Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Laurent Monasse, Virginie Daru, Christian Mariotti, and Serge Piperno A Preconditioned Euler Flow Solver for Simulation of Helicopter Rotor Flow in Hover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 Kazem Hejranfar and Ramin Kamali Moghadam A New Type of Gas-Kinetic Upwind Euler/N-S Solvers . . . . . . . . . . . . . . . . . 485 Lei Tang Part XIX High-Order Accuracy Efficiency Enhancement in High Order Accurate Euler Computation via AWM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495 Hyungmin Kang, Dongho Lee, Dohyung Lee, Dochan Kwak, and John Seo On the Construction of High Order Finite Volume Methods . . . . . . . . . . . . 501 Wanai Li, Guodong Lei, and Yu-xin Ren Multiblock Computations of Complex Turbomachinery Flows Using Residual-Based Compact Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 Bertrand Michel, Paola Cinnella, and Alain Lerat A Local Resolution Refinement Algorithm Using Gauss–Lobatto Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 J.S. Shang and P.G. Huang

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Performance Comparison of High Resolution Schemes . . . . . . . . . . . . . . . . . 527 Müslüm Arıcı and Hüseyin Sinasi ¸ Onur Solving the Convective Transport Equation with Several High-Resolution Finite Volume Schemes: Test Computations . . . . . . . . . . . . 535 Alexander I. Khrabry, Evgueni M. Smirnov, and Dmitry K. Zaytsev Part XX Free-Surface Flow Computational Study of Hydrodynamics and Heat Transfer Associated with a Liquid Drop Impacting a Hot Surface . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Edin Berberović, Ilia V. Roisman, Suad Jakirlić, and Cameron Tropea A Second Order JFNK-Based IMEX Method for Single and Multi-Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Samet Kadioglu, Dana Knoll, Mark Sussman, and Richard Martineau Simulation of Two-Phase Flow in Sloshing Tanks . . . . . . . . . . . . . . . . . . . . . . 555 Roel Luppes, Arthur Veldman, and Rik Wemmenhove High-Precision Reconstruction of Gas-Liquid Interface in PLIC-VOF Framework on Unstructured Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Kei Ito, Tomoaki Kunugi, and Hiroyuki Ohshima Part XXI Mesh Adaptation Adjoint-Based Methodology for Anisotropic Grid Adaptation . . . . . . . . . . . 571 Nail K. Yamaleev, Boris Diskin, and Kedar Pathak Transient Adaptive Algorithm Based on Residual Error Estimation . . . . . 577 N. Ganesh and N. Balakrishnan Adaptive and Consistent Properties Reconstruction for Complex Fluids Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 Guoping Xia, Chenzhou Lian, and Charles L. Merkle Space-Filling Curve Techniques for Parallel, Multiscale-Based Grid Adaptation: Concepts and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591 Sorana Melian, Kolja Brix, Siegfried Müller, and Gero Schieffer

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Part XXII Immersed Boundary Method Recent Advances in the Development of an Immersed Boundary Method for Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 M.D. de Tullio, P. De Palma, M. Napolitano, and G. Pascazio An Overview of the LS-STAG Immersed Boundary Method for Viscous Incompressible Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607 Olivier Botella and Yoann Cheny A Two-Dimensional Embedded-Boundary Method for Convection Problems with Moving Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613 Yunus Hassen and Barry Koren A Second-Order Immersed Boundary Method for the Numerical Simulation of Two-Dimensional Incompressible Viscous Flows Past Obstacles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 621 François Bouchon, Thierry Dubois, and Nicolas James Part XXIII Gas-Kinetic BGK Schemes A New High-Order Multidimensional Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 629 Qibing Li, Kun Xu, and Song Fu A Direct Boltzmann-BGK Equation Solver for Arbitrary Statistics Using the Conservation Element/Solution Element and Discrete Ordinate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637 Bagus Putra Muljadi and Jaw-Yen Yang Part XXIV Extreme Flows Space-Time Convergence Analysis of a Dual-Time Stepping Method for Simulating Ignition Overpressure Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645 Jeffrey A. Housman, Michael F. Barad, Cetin C. Kiris, and Dochan Kwak Numerical Solution of Maximum-Entropy-Based Hyperbolic Moment Closures for the Prediction of Viscous Heat-Conducting Gaseous Flows . . 653 James G. McDonald and Clinton P.T. Groth Supercritical-Fluid Flow Simulations Across Critical Point . . . . . . . . . . . . . 661 Satoru Yamamoto, Takashi Furusawa, and Ryo Matsuzawa

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Numerical Model for the Analysis of the Thermal-Hydraulic Behaviors in the Calandria Based Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Mula Venkata Ramana Reddy, S.D. Ravi, P.S. Kulkarni, and N.K.S. Rajan Part XXV Deformable Bodies Seamless Virtual Boundary Method for Complicated Incompressible Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679 Hidetoshi Nishida Modeling, Simulation and Control of Fish-Like Swimming . . . . . . . . . . . . . 689 Michel Bergmann and Angelo Iollo Multi-Level Quasi-Newton Methods for the Partitioned Simulation of Fluid-Structure Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695 Joris Degroote, Sebastiaan Annerel, and Jan Vierendeels Entropic Lattice Boltzmann Simulation for Unsteady Flow Around two Square Cylinders Arranged Side by Side in a Channel . . . . . . . . . . . . . . . . . . 701 Takahiro Yasuda and Nobuyuki Satofuka An Eulerian Approach for Fluid and Elastic Structure Interaction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707 Koji Morinishi and Tomohiro Fukui Part XXVI DNS/RANS Computations Dissipation Element Analysis of Inhomogenous Turbulence . . . . . . . . . . . . . 717 Philip Schaefer, Markus Gampert, Jens Henrik Goebbert, and Norbert Peters A Matrix-Free Viscous Linearization Procedure for Implicit Compressible Flow Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723 Nikhil Vijay Shende and N. Balakrishnan DNS of Shock/Boundary Layer Interaction Flow in a Supersonic Compression Ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729 Xin-Liang Li, De-Xun Fu, Yan-Wen Ma, and Xian Liang An Efficient Generator of Synthetic Turbulence at RANS–LES Interface in Embedded LES of Wall-Bounded and Free Shear Flows . . . . . . . . . . . . . . 739 Dmitry Adamian and Andrey Travin

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A Numerical Simulation of the Magnetically Driven Flows in a Square Container Using the Delayed Detached Eddy Simulation . . . . . . . . . . . . . . . 745 Karel Fraˇna, Vít Honzejk, and Kateˇrina Horáková On the Performance of Optimized Finite Difference Schemes in Large-Eddy Simulation of the Taylor–Green Vortex . . . . . . . . . . . . . . . . . . 753 Dieter Fauconnier and Erik Dick

Part XXVII Combustion/Supersonic Flow Wide-Range Single Engine Operated from Subsonic to Hypersonic Conditions: Designed by Computational Fluid Dynamics . . . . . . . . . . . . . . . 763 Ken Naitoh, Kazushi Nakamura, Takehiro Emoto, and Takafumi Shimada Numerical Simulations of the Performance of Scramjet Engine Model with Pylon Set Located in the Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769 I.M. Blankson, A.L. Gonor, and V.A. Khaikine Scale Adaptive Simulations over a Supersonic Car . . . . . . . . . . . . . . . . . . . . . 779 Guillermo Araya, Ben Evans, Oubay Hassan, and Kenneth Morgan Rotating Detonation Engine Injection Velocity Limit and Nozzle Effects on Its Propulsion Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789 Jian-Ping Wang and Ye-Tao Shao

Part XXVIII Mesh Adaptation/Decomposition Singularities in Lid Driven Cavity Solved by Adjusted Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 799 Pavel Burda, Jaroslav Novotný, and Jakub Šístek A Parallel Implementation of the BDDC Method for the Stokes Flow . . . . . 807 Jakub Šístek, Pavel Burda, Jan Mandel, Jaroslav Novotný, and Bedˇrich Sousedík An Adaptive Multiwavelet-Based DG Discretization for Compressible Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813 Francesca Iacono, Georg May, Siegfried Müller, and Roland Schäfer

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Part XXIX Micro- and Nano-Fluidics Massively Parallel Mesoscopic Simulations of Gas Permeability of Thin Films Composed of Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823 Alexey N. Volkov and Leonid V. Zhigilei Predicting Three-Dimensional Inertial Thin Film Flow over Micro-Scale Topography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833 Sergii Veremieiev, Philip H. Gaskell, Yeaw Chu Lee, and Harvey M. Thompson Part XXX Level Set Method A Parallel Adaptive Projection Method for Incompressible Two Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 841 Davide Zuzio and Jean-Luc Estivalezes Hybrid Particle Level-Set Method for Convection-Diffusion Problems in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847 Héloïse Beaugendre, S. Huberson, and I. Mortazavi Part XXXI Technical Notes Numerical Solution of Boundary Control Problems for Boussinesq Model of Heat Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857 Gennady Alekseev and Dmitry Tereshko Blade Design Effects on the Performance of a Centrifugal Pump Impeller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859 John S. Anagnostopoulos The Local Flux Correction Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863 Bolat M. Baimirov Adaptive Meshes to Improve the Linear Stability Analysis of the Flow Past a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 867 Iago C. Barbeiro, Julio R. Meneghini, and J.A.P. Aranha Migration of Species into a Particle Under Different Flow Conditions . . . . 869 Alfredo R. Carella and Carlos A. Dorao

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On the Role of Numerical Dissipation in Unsteady Low Mach Number Flow Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 Yann Moguen, Tarik Kousksou, Erik Dick, and Pascal Bruel Unsteady Flow Around Two Tandem Cylinders Using Advanced Turbulence Modeling Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 879 Jian Liu, Zhixiang Xiao, and Song Fu High Order Versions of the Collocations and Least Squares Method for Navier–Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883 Vadim Isaev and Vasily Shapeev Large-Eddy Simulation of the Flow over a Thin Airfoil at Low Reynolds Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885 Ryoji Kojima, Taku Nonomura, Akira Oyama, and Kozo Fujii Adverse Free-Stream Conditions for Transonic Airfoils with Concave Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 887 Alexander Kuzmin Development of a Conservative Overset Mesh Method on Unstructured Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893 Mun Seung Jung and Oh Joon Kwon Symmetrical Vortex Fragmenton as a Vortex Element for Incompressible 3D Flow Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 I.K. Marchevsky and G.A. Scheglov Morphogenic Computational Fluid Dynamics: Brain Shape Similar to the Engine Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 Haruka Kawanobe, Shou Kimura, and Ken Naitoh Computational Modeling of the Flow and Noise for 3-D Exhaust Turbulent Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903 S.A. Cheprasov, D.A. Lyubimov, A.N. Secundov, K. Ya. Yakubovsky, and S.F. Birch Numerical Simulation of Laser Welding of Thin Metallic Plates Taking into Account Convection in the Welding Pool . . . . . . . . . . . . . . . . . . . . . . . . . . 909 Vasily Shapeev, Vadim Isaev, and Anatoly Cherepanov Numerical Simulation of Cavitation Bubble Collapse Near Wall . . . . . . . . . 913 Byeong Rog Shin

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Numerical Analysis of Airflow in Human Vocal Folds Using Finite Element and Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 917 Petr Šidlof, Bernhard Müller, and Jaromír Horáˇcek Part XXXII Appendix: Papers Presented in ICCFD5 but Unpublished in “Computational Fluid Dynamics 2008” Analysis of Free-Shear Flow Noise Through a Decomposition of the Lighthill Source Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923 Florent Margnat, Véronique Fortuné, Peter Jordan, and Yves Gervais Thermal Reaction Wave Simulation Using Micro and Macro Scale Interaction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929 Andrey Markov, Igor Filimonov, and Karen Martirosyan CFD Study on Flow Field inside B.I.P.C. Steam Boilers . . . . . . . . . . . . . . . . . 937 F. Panahi Zadeh and M. Farzaneh Gord Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 941

Contributors

Dmitry Adamian St. Petersburg State Polytechnic University, St. Petersburg 195257, Russia, [email protected] L. Afferrante CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy, [email protected] Gennady Alekseev Institute of Applied Mathematics FEB RAS, 7, Vladivostok 690041, Russia, [email protected] John S. Anagnostopoulos School of Mechanical Engineering, National Technical University of Athens, Athens, Greece, [email protected] Sebastiaan Annerel Department of Flow, Heat and Combustion Mechanics, Ghent University, 9000 Ghent, Belgium, [email protected] J.A.P. Aranha NDF, Escola Politècnica, University of Säo Paulo, Säo Paulo, Brazil, [email protected] Guillermo Araya Civil & Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK, [email protected] Müslüm Arıcı Engineering Faculty, Department of Mechanical Engineering, Kocaeli University, 41380 Kocaeli, Turkey, [email protected] Bolat M. Baimirov Kazakh National University, Almaty, Kazakhstan, [email protected] N. Balakrishnan Computational Aerodynamics Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India, [email protected] Michael F. Barad University of California, Davis, CA 95616, USA, [email protected] Iago C. Barbeiro NDF, Escola Politécnica, University of Säo Paulo, Säo Paulo, Brazil, [email protected] B. Basara AVL LIST GmbH, Advanced Simulation Technologies, 8020 Graz, Austria, [email protected] xxi

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Contributors

Héloïse Beaugendre IMB, Institut Polytechnique de Bordeaux and INRIA Bordeaux Sud-Ouest projet MC2, 351 Cours de La Libération, 33405 Talence Cedex, France, [email protected] Edin Berberović Chair for Energy Technology and Process Engineering, University of Zenica, 72000 Zenica, Bosnia and Herzegovina, [email protected] Michel Bergmann INRIA Bordeaux Sud Ouest, Team MC2, Bordeaux F-33000, France; Institut de Mathmatiques de Bordeaux, UMR 5251, Bordeaux F-33000, France, [email protected] Germain Billet ONERA, BP72 FR-92322, Chatillon Cedex, France, [email protected] S.F. Birch The Boeing Company, Seattle WA 98124, USA, [email protected]; [email protected] I.M. Blankson NASA Glenn Research Center, Cleveland, OH 44135, USA, [email protected] Andrzej Boguslawski Czestochowa University of Technology, 42-200 Czestochowa, Poland, [email protected] Michel Borrel ONERA, FR-92322, Chatillon Cedex, France, [email protected] Olivier Botella LEMTA, Nancy-University, CNRS, 54504 Vandœuvre, France, [email protected] François Bouchon Laboratoire de Mathématiques, Université Blaise Pascal and CNRS (UMR 6620), Campus Universitaire des Cézeaux, 63177 Aubiere, France, [email protected] Didier Bresch Laboratoire de Mathématiques, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac, France, [email protected] Kolja Brix Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany, [email protected] Pascal Bruel Laboratoire de Mathèmatiques et de leurs, CNRS and Universitè de Pau et des Pays de ÍAdour, Applications, UMR 5142 CNRS-UPPA, BP 1155 - 64 013 Pau, France, [email protected] C.-H. Bruneau IMB, UB1, Université de Bordeaux, Team MC2 INRIA Bordeaux-Sud-Ouest 351 cours de la Libération, F-33405 Talence, France, [email protected] Pavel Burda Faculty of Mechanical Engineering, Department of Mathematics, Czech Technical University, CZ-121 35 Praha 2, Czech Republic, [email protected] Alfredo R. Carella Norwegian University of Science and Technology, N-7491 Trondheim, Norway, [email protected]

Contributors

xxiii

M.H. Carpenter NASA Langley Research Center, Hampton, VA, USA, [email protected] Gregoire Casalis ONERA Toulouse Cedex 4, France, [email protected] Jean-Jacques Chattot Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616-5294, USA, [email protected] Yoann Cheny CERFACS, 31057 Toulouse Cedex 01, France, [email protected] S.A. Cheprasov Scientific and Research Center ECOLEN, 111116 Moscow, Russia, [email protected] Anatoly Cherepanov Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia, [email protected] Oussama Chikhaoui Département d’Aérodynamique, Énergétique et Propulsion, Institut Supérieur de l’Aéronautique et de l’Espace, 10 av. Edouard Belin - BP 54032 – 31055, Toulouse, France, [email protected] Vincent H. Chu Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, Canada, [email protected] Paola Cinnella Laboratoire DynFluid, Arts et Mètiers–ParisTech, 75013 Paris, France, [email protected] E. Creusé Université Lille 1, Team SIMPAF INRIA Lille Nord Europe, UMR 8524 CNRS Cité Scientifique, F-59655 Villeneuve d’Ascq, France, [email protected] Virginie Daru LIMSI - CNRS, 91403 Orsay, France; Arts et Métiers Paris Tech, DynFluid Lab, 75013 Paris, France, [email protected] P. De Palma CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy, [email protected] M.D. de Tullio CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy, [email protected] Joris Degroote Department of Flow, Heat and Combustion Mechanics, Ghent University, 9000 Ghent, Belgium, [email protected] D. Depeyras IMB, UB1, Université de Bordeaux, Team MC2 INRIA Bordeaux-Sud-Ouest 351 cours de la Libération, F-33405 Talence, France, [email protected] Erik Dick Faculty of Engineering, Department of Flow, Heat and Combustion Mechanics, Ghent University, 9000 Ghent, Belgium, [email protected] Boris Diskin National Institute of Aerospace, Hampton, VA, USA, [email protected]

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Contributors

Carlos A. Dorao Department of Energy and Process Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway, [email protected] Kateryna Dorogan EDF R&D, MFEE, F-78400 Chatou, France; LATP, CMI, 13453 Marseille, France, [email protected] Thierry Dubois Laboratoire de Mathematiques, Universite Blaise Pascal and CNRS (UMR 6620), Campus Universitaire des Cezeaux, 63177 Aubiere, France, [email protected] Michael Dumbser Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering, University of Trento I-38100 Trento TN, Italy; Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, D70550 Stuttgart, Germany, [email protected], [email protected] Takehiro Emoto Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan, [email protected] Vahid Esfahanian Faculty of Engineering, Department of Mechanical Engineering, University of Tehran, Tehran, Iran, [email protected] Jean-Luc Estivalezes Department of Models For Aerodynamics and Energy, ONERA, Toulouse, France, [email protected] Ben Evans Civil & Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK, [email protected] Sinan Eyi Middle East Technical University, 06531 Ankara, Turkey, [email protected] Alper Ezertas Middle East Technical University, 06531 Ankara, Turkey, [email protected] Dieter Fauconnier Department of Flow, Heat and Combustion Mechanics, Faculty of Engineering, Ghent University, 9000 Ghent, Belgium, [email protected] O. Feodoritova Keldysh Institute of Applied Mathematics, Moscow, Russia, [email protected] Enrique D. Fernandez-Nieto Departamento de Matemática Aplicada I, E.T.S. Arquitectura, Universidad de Sevilla, 41012 Sevilla, Spain, [email protected] Igor Filimonov Institute of Structural Macrokinetics and Material Science of the Russian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia, [email protected] Véronique Fortuné Laboratoires d’Etudes Aèrodynamiques, UMR CNRS 6609, ENSMA, Universitè de POITIERS, 86022 Poitiers Cedex, France, [email protected]

Contributors

xxv

Y. Fraigneau LIMSI – UPR 3251 CNRS, BP.133, 91403 ORSAY Cedex, France, [email protected] Erwin Franquet LaTEP – ENSGTI, rue Jules Ferry, BP 7511, 64075 Paucedex, France, [email protected] ˇ Department of Power Engineering Equipment, Technical University Karel Frana of Liberec, 461 17 Liberec, Czech Republic, [email protected] De-xun Fu LHD, Institute of Mechanics, CAS, Beijing 100190, China, [email protected] Song Fu Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China; School of Aerospace Engineering, Tsinghua University, Beijing, China, [email protected] Kozo Fujii Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Sagamihara, Kanagawa, Japan, [email protected] Tomohiro Fukui Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan, [email protected] Takashi Furusawa Department of Computer and Mathematical Sciences, Tohoku University, Sendai 980-8579, Japan, [email protected] A.M. Gambaruto Department of Mathematics and CEMAT, Instituto Superior Técnico, Lisbon, Portugal, [email protected] Markus Gampert Institut für Technische Verbrennung, RWTH Aachen, 52056 Aachen, Germany, [email protected] N. Ganesh St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN, USA, [email protected] Haiyang Gao Department of Aerospace Engineering, CFD Center, Iowa State University, Ames, IA 50011, USA, [email protected] Andrey V. Garbaruk St. Petersburg Polytechnic University, St. Petersburg 195257, Russia, [email protected] Philip H. Gaskell School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK, [email protected] Gregor Gassner Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, D70550 Stuttgart, Germany, [email protected] Yves Gervais Laboratoires d’Etudes Aèrodynamiques, UMR CNRS 6609, ENSMA, Universitè de Poitiers, 86022 Poitiers Cedex, France, [email protected] P. Gilliéron Technocentre Renault, Direction de La Recherche, DREAM/DTAA 1, F-78288 Guyancourt, France, [email protected]

xxvi

Contributors

S.K. Godunov Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia, [email protected] Jens Henrik Goebbert Institut für Technische Verbrennung, RWTH Aachen, 52056 Aachen, Germany, [email protected] A.L. Gonor Hampton University, Hampton, VA 23668, USA, [email protected] L.F. Gonzalez School of Engineering System, Queensland University of Technology, Brisbane, Australia, [email protected] M. Farzaneh Gord Shahrood University of Technology, Shahrood, Iran, [email protected] Jérémie Gressier Département d’Aérodynamique, Énergétique et Propulsion, Institut Supérieur de l’Aéronautique et de l’Espace, Toulouse, France, [email protected] Gilles Grondin Département d’Aérodynamique, Énergétique et Propulsion, Institut Supérieur de l’Aéronautique et de l’Espace, Toulouse, France, [email protected] Clinton P.T. Groth University of Toronto, Institute for Aerospace Studies, Toronto ON, Canada, M3H 5T6, [email protected] Marshall R. Gusman ELORET Corp., Sunnyvale, CA 94086, USA, [email protected] Mohamed M. Hafez Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616-5294, USA, [email protected] Takanori Haga Department of Aerospace Engineering, CFD Center, Iowa State University, Ames, IA 50011, USA, [email protected] Laurence Halpern LAGA, Université Paris XIII and CNRS, 93430 Villetaneuse, France, [email protected] Oubay Hassan Civil & Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK, [email protected] Yunus Hassen Centrum Wiskunde & Informatica, Amsterdam, The Netherlands; Faculty of Aerospace Engineering, TU Delft, The Netherlands, [email protected] Kazem Hejranfar Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran, [email protected] Jean-Marc Hérard EDF R&D, MFEE, F-78400 Chatou, [email protected] Christian Heschl Fachhochschulstudiengänge Burgenland - University of Applied Science, Pinkafeld, Austria, [email protected]

Contributors

xxvii

Jason E. Hicken University of Toronto, Institute for Aerospace Studies, Toronto, ON, Canada M3H 5T6, [email protected] Florian Hindenlang Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, D70550 Stuttgart, Germany, [email protected] Vít Honzejk Department of Power Engineering Equipment, Technical University of Liberec, 461 17, Liberec, Czech Republic, [email protected] Jaromír Horáˇcek Institute of Thermomechanics, Academy of Sciences of the Czech Republic, 182 00 Prague 8, Czech Republic, [email protected] Kateˇrina Horáková Department of Power Engineering Equipment, Technical University of Liberec, 461 17, Liberec, Czech Republic, [email protected] Jeffrey A. Housman ELORET Corp., Sunnyvale, CA 94086, USA, [email protected] P.G. Huang Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH, USA, [email protected] Jingbo Huang School of Aerospace Engineering, Tsinghua University, Beijing, China, [email protected] S. Huberson Institut Prime, Universite Poitiers-ENSMA, 86962 Futuroscope Chaseneuil Cedex, France, [email protected] Francesca Iacono AICES, RWTH Aachen, 52056 Aachen, Germany, [email protected] Shigeru Ikeo Sophia University, Tokyo 102-8554 Japan, [email protected] Kiao Inthavong RMIT University, Bundoora, Australia, [email protected] Angelo Iollo INRIA Bordeaux Sud Ouest, Team MC2, Bordeaux F-33000, France; Institut de Mathmatiques de Bordeaux, UMR 5251, Bordeaux F-33000, France, [email protected] Ioan Ionescu Laboratoire de Mathématiques, UMR 5127 CNRS, Université de Savoie, 73376 Le Bourget du Lac, France, [email protected] Vadim Isaev Novosibirsk State University, Novosibirsk 630090, Russia, [email protected] Kei Ito Japan Atomic Energy Agency, Ibaraki 311-1393, Japan, [email protected] Suad Jakirlić Institute of Fluid Mechanics and Aerodynamics/Center of Smart Interfaces, Technische Universität Darmstadt, D-64287 Darmstadt, Germany, [email protected]

xxviii

Contributors

H.A. Jakobsen Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway, [email protected] Nicolas James Laboratoire de Mathematiques, Universite Blaise Pascal and CNRS (UMR 6620), Campus Universitaire des Cezeaux, 63177 Aubiere, France, [email protected] Antony Jameson Stanford University, Palo Alto, CA 94305, USA, [email protected] Peter Jordan Laboratoires d’Etudes Aèrodynamiques, UMR CNRS 6609, ENSMA, Universitè de POITIERS, 86022 Poitiers Cedex, France, [email protected] Mun Seung Jung Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, [email protected] Samet Kadioglu Idaho National Laboratory, Fuels Modeling and Simulation Department, Idaho Falls, ID 83415, USA, [email protected] D. Kamenetskii Boeing Commercial Airplanes, Seattle, WA, USA, [email protected] Sean J. Kamkar Stanford University, Palo Alto, CA 94305, USA, [email protected] Hyungmin Kang School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, S. Korea, [email protected] Aaron Katz US Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA 94035, USA, [email protected] Haruka Kawanobe Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan, [email protected] V.A. Khaikine Hampton University, Hampton, VA 23668, USA, [email protected] Alexander I. Khrabry St. Petersburg State Polytechnic University, St. Petersburg 195251, Russia, [email protected] Chongam Kim School of Mechanical and Aerospace Engineering, Institute of Advanced Aerospace Technology, Seoul National University, Seoul 151-744, Korea, [email protected] Shou Kimura Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan, [email protected] Cetin C. Kiris NASA Ames Research Center, Moffett Field, CA 94035, USA, [email protected]

Contributors

xxix

Dana Knoll Los Alamos National Laboratory, Theoretical Division (T-3), Los Alamos, NM 87545, USA, [email protected] Ryoji Kojima Department of Aeronautics and Astronautics, University of Tokyo, Sagamihara, Kanagawa 252-5210, Japan, [email protected] V. Koláˇr Institute of Hydrodynamics, Academy of Sciences of the Czech Republic, CZ-16612 Prague 6, Czech Republic, [email protected] Barry Koren Centrum Wiskunde & Informatica, Amsterdam, The Netherlands; Faculty of Aerospace Engineering, TU Delft, The Netherlands, [email protected] Tarik Kousksou Laboratoire de Thermique, Universitè de Pau et des Pays de ÍAdour, Énergètique et Procèdès, ENSGTI, 64 075 Pau, France, [email protected] Sergey V. Kravchenko The Boeing Company, Chicago, IL 60606-1596, USA, [email protected] P.S. Kulkarni Computational Mechanics Laboratory, JATP, Deptartment of Aerospace Enggineering, Indian Institute of Science (IISc), Bangalore, India, [email protected] Tomoaki Kunugi Kyoto University, Kyoto 606-8501, Japan, [email protected] Alexander Kuzmin Laboratory of Aerodynamics, St. Petersburg State University, St. Petersburg 198504, Russia, [email protected] Dochan Kwak NAS Applications Branch, NASA Ames Research Center, Moffett Field, CA 94035-1000, USA, [email protected] Oh Joon Kwon Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea, [email protected] C.J. Lee School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia, [email protected] Dohyung Lee Department of Mechanical Engineering, Hanyang University, Ansan, S. Korea, [email protected] Dongho Lee School of Mechanical and Aerospace Engineering, Seoul National University, Seoul, S. Korea, [email protected] D.S. Lee International Center for Numerical Methods in Engineering (CIMNE/UPC), Barcelona, Spain; School of Aerospace, Mechanical And Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia, [email protected] Yeaw Chu Lee School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK, [email protected]

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Contributors

Guodong Lei Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China, [email protected] Alain Lerat Laboratoire DynFluid, Arts et Métiers-ParisTech, 75013, Paris, France, [email protected] Qibing Li Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China, [email protected] Wanai Li Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China, [email protected] Xin-Liang Li LHD, Institute of Mechanics, CAS, Beijing 100190, China, [email protected] Chenzhou Lian Purdue University, West Lafayette, IN 47907, USA, [email protected] Xian Liang LHD, Institute of Mechanics, CAS, Beijing 100190, China, [email protected] Jian Liu School of Aerospace Engineering, Tsinghua University, 100084, Beijing, P.R. China, [email protected] Roel Luppes J. Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 9700 AK Groningen, The Netherlands, [email protected] D.A. Lyubimov Scientific and Research Center ECOLEN, Moscow 111116, Russia, [email protected] Yan-Wen Ma LHD, Institute of Mechanics, CAS, Beijing 100190, China, [email protected] Robert Maduta Institute of Fluid Mechanics and Aerodynamics, Technische Universität Darmstadt, D-64287 Darmstadt, Germany, [email protected] Jan Mandel Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80217-3364, USA, [email protected] I.K. Marchevsky Bauman Moscow State Technical University, Moscow 105005, Russia, [email protected] Maciej Marek Czestochowa University of Technology, 42-200 Czestochowa, Poland, [email protected] Florent Margnat Laboratoire SINUMEF, Arts & Métiers ParisTech, 151 boulevard de l’Hôpital, 75013 Paris, France, [email protected] Christian Mariotti CEA DIF, F-91297 Arpajon, France, [email protected] Andrey Markov Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow 119526, Russia, [email protected]

Contributors

xxxi

Richard Martineau Idaho National Laboratory, Fuels Modeling and Simulation Department, Idaho Falls, ID 83415, USA, [email protected] Karen Martirosyan Department of Physics and Astronomy, University of Texas at Brownsville, Brownsville, TX 78520, USA, [email protected] A. Martynov Keldysh Institute of Applied Mathematics, Moscow, Russia, [email protected] Ryo Matsuzawa Department of Computer and Mathematical Sciences, Tohoku University, Sendai 980-8579, Japan, [email protected] Dimitri J. Mavriplis Depratment of Mechanical Engineering, University of Wyoming, Laramie 82071, USA, [email protected] Georg May AICES Institute for Advanced Study in Computation Engineering Science, RWTH Aachen University, 52056 Aachen, Germany, [email protected] James G. McDonald University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada M3H 5T6, [email protected] S. Medvedev Keldysh Institute of Applied Mathematics, Moscow, Russia, [email protected] Sorana Melian Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany, [email protected] Julio R. Meneghini NDF, Escola Politècnica, University of Säo Paulo, Säo Paulo, Brazil, [email protected] Charles L. Merkle Purdue University, West Lafayette, IN 47907, USA, [email protected] Fabien Mery ONERA, BP 4025-31055 Toulouse Cedex 4, France, [email protected] Bertrand Michel Département DSNA ONERA, 92320 Châtillon, France, [email protected] Jean-Pierre Minier EDF R&D, MFEE, F-78400 Chatou [email protected] Charles Mockett Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, 10623 Berlin, Germany, [email protected] Ramin Kamali Moghadam Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran, [email protected] Yann Moguen Department of Flow, Heat and Combustion, Ghent University, Sint-Pietersnieuwstraat, 41, 9000 Gent, Belgium, [email protected]

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Contributors

Laurent Monasse Université Paris-Est, CERMICS, 77455 Marne-la-Vallée, France, [email protected] Kenneth Morgan Civil & Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK, [email protected] Koji Morinishi Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan, [email protected] I. Mortazavi IMB, UB1, Université de Bordeaux, Team MC2 INRIA Bordeaux-Sud-Ouest 351 cours de la Libération, F-33405 Talence, France, [email protected] P. Moses Faculty of Mechanical Engineering, Department of Mathematics, Czech Technical University in Prague, CZ-12135 Prague 2, Czech Republic, [email protected] Bagus Putra Muljadi Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan, [email protected] Bernhard Müller Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim NO-7491, Norway, [email protected] Siegfried Müller Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany, [email protected] Claus-Dieter Munz Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, D70550 Stuttgart, Germany, [email protected] Ken Naitoh Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan, [email protected] Mehdi Najafi Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran, [email protected] Kazuhiro Nakahashi Department of Aerospace Engineering, Tohoku University, Sendai 980-8579, Japan, [email protected] Kazushi Nakamura Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan, [email protected] M. Napolitano CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy, [email protected] Hidetoshi Nishida Department of Mechanical and System Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan, [email protected] Boniface Nkonga INRIA PUMAS, Université de Nice, France, [email protected] Sebastian Noelle IGPM, RWTH Aachen University, 52062 Aachen, Germany, [email protected]

Contributors

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Taku Nonomura Institute of Space and Astronautical Science, Japan; Aerospace Exploration Agency (JAXA), Sagamihara, Japan, [email protected] Jaroslav Novotný Institute of Thermomechanics, Academy of Sciences of the Czech Republic, CZ-182 00 Praha 8, Czech Republic; Faculty of Civil Engineering, Department of Mathemetics, Czech Technical University in Prague, CZ-166 29 Praha 6, Czech Republic, [email protected]; [email protected] Yasuo Obikane Institute of Computational Fluid Dynamics, Meguroku-ku, Tokyo 152-0011, Japan; Department of Mechanical Engineering, Sophia University, Tokyo 102-8554, Japan, [email protected]; [email protected] Gerard M. O’Connor School of Physics, National University of Ireland Galway, Galway, Ireland, [email protected] Hiroyuki Ohshima Japan Atomic Energy Agency, Ibaraki 311-1393, Japan, [email protected] E. Onate International Center for Numerical Methods in Engineering (CIMNE/UPC), Barcelona, Spain, [email protected] Hüseyin Sinasi ¸ Onur Engineering Faculty, Department of Mechanical Engineering, Kocaeli University, 41380 Kocaeli, Turkey, [email protected] Michal Osusky University of Toronto, Institute for Aerospace Studies, Toronto, ON, Canada M3H 5T6, [email protected] Akira Oyama Institute of Space and Astronautical Science, Japan; Aerospace Exploration Agency, Sagamihara, Kanagawa, Japan, [email protected] Jin Seok Park School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-744, Korea, [email protected] G. Pascazio CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy, [email protected] Kedar Pathak North Carolina A&T State University, Greensboro, NC, USA, [email protected] Z. Pavlovic AVL LIST GmbH, Advanced Simulation Technologies, SI-2000 Maribor, Slovenia, [email protected] J. Periaux International Center for Numerical Methods in Engineering (CIMNE/UPC), Barcelona, Spain; Department of Mathematical Information Technology, University of Jyvaskyla, Jyvaskyla FI-40014, Finland, [email protected] Vincent Perrier INRIA Bordeaux Sud Ouest and Laboratoire de Mathématiques et de leur Applications, Bâtiment IPRA, Université de Pau et des Pays de l’Adour, Avenue de l’Université, 64013 Pau Cedex, France, [email protected] Norbert Peters Institut für Technische Verbrennung, RWTH Aachen, 52056 Aachen, Germany, [email protected]

xxxiv

Contributors

Serge Piperno Ecole des Ponts ParisTech, 77455 Marne-la-Vallée, France, [email protected] N. Qin Department of Mechanical Engineering, University of Sheffield, Sheffield, UK, [email protected] N.K.S. Rajan CGPL, Department of Aerospace Engineeering, IISc, Bangalore, Indian Institute of Science (IISc), Bangalore, India, [email protected] S.D. Ravi CGPL, Department of Aerospace Engineering, Indian Institute of Science (IISc), Bangalore, India, [email protected] Yu-xin Ren Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China, [email protected] Andreas Richter Institute for Aerospace Engineering, Technische Universität Dresden, 01062 Dresden, Germany, [email protected] Philip L. Roe University of Michigan, Ann Arbor, MI, USA, [email protected] Ilia V. Roisman Chair of Fluid Mechanics and Aerodynamics/Center of Smart Interfaces, Technische Universität Darmstadt, 64287 Darmstadt, Germany, [email protected] Markus P. Rumpfkeil Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071, USA, [email protected] Juliette Ryan ONERA, BP72 – 29 avenue de la Division Leclerc, FR-92322, Chatillon Cedex, France, [email protected] Ryotaro Sakai Department of Aerospace Engineering, Tohoku University, Sendai 980-8579, Japan, [email protected] Manuel D. Salas NASA Langley Research Center, Hampton, VA 23681, USA, [email protected] Venkateswaran Sankaran US Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA 94035, USA, [email protected] Daisuke Sasaki Department of Aerospace Engineering, Tohoku University, Sendai 980-8579, Japan, [email protected] Nobuyuki Satofuka The University of Shiga Prefecture, 2500 Hassaka, Hikone, Japan, [email protected] Philip Schaefer Institut für Technische Verbrennung, RWTH Aachen, 52056 Aachen, Germany, [email protected] Roland Schäfer IGPM, RWTH Aachen, 52056 Aachen, Germany, [email protected] G.A. Scheglov Bauman Moscow State Technical University, Moscow 105005, Russia, [email protected]

Contributors

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Gero Schieffer Lehrstuhl für Computergestützte Analyse Technischer Systeme, RWTH Aachen University, 52052, Aachen, Germany, [email protected] Sven Schmitz University of California, Davis, CA, USA, [email protected] David M. Schuster NASA Engineering and Safety Center, NASA Langley Research Center, Hampton, VA 23681, USA, [email protected] Jochen Schütz AICES Institute for Advanced Study in Computation Engineering Science, RWTH Aachen University, 52062 Aachen, Germany, [email protected] A.N. Secundov Scientific and Research Center ECOLEN, Moscow 111116, Russia, [email protected] John Seo Asian Office of Aerospace Research and Development, Air Force Office of Scientific Research, Tokyo, Japan, [email protected] A. Sequeira Department of Mathematics and CEMAT, Instituto Superior Técnico, Lisbon, Portugal, [email protected] J.S. Shang Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH, USA, [email protected] Ye-Tao Shao State Key Laboratory of Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, China, [email protected] Vasily Shapeev Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia, [email protected] Nikhil Vijay Shende Computational Aerodynamics Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India, [email protected] Takafumi Shimada Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169–8555, Japan, [email protected] Byeong Rog Shin Department of Mechanical Engineering, Changwon National University, Changwon 641-773, Korea, [email protected] Mikhail L. Shur New Technologies and Services and State Polytechnic University, St. Petersburg 195220, Russia, [email protected] Petr Šidlof Institute of Thermomechanics, Academy of Sciences of the Czech Republic, 182 00 Prague 8, Czech Republic, [email protected] Jakub Šístek Institute of Mathematics, Academy of Sciences of the Czech Republic, CZ-11567 Prague 1, Czech Republic, [email protected]

xxxvi

Contributors

Evgueni M. Smirnov St. Petersburg State Polytechnic University, St. Petersburg 195251, Russia, [email protected] Bedˇrich Sousedík Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80217-3364, USA, [email protected] Philippe R. Spalart Boeing Commercial Airplanes, Seattle, WA 98124, USA, [email protected] F. Sporleder Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway, [email protected] K. Srinivas School of Aerospace, Mechanical And Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia, [email protected] Jörg Stiller Institute for Aerospace Engineering, Technische Universität Dresden, 01062 Dresden, Germany, [email protected] Mikhail Kh. Strelets New Technologies & Services and St. Petersburg State Polytechnic University, St. Petersburg 195220, Russia, [email protected] Mark Sussman Department of Mathematics, Florida State University Tallahassee, FL 32306, USA, [email protected] Kensuke Suzuki University of California, Davis, CA, USA, [email protected] R.C. Swanson NASA Langley Research Center, Hampton, VA 23681, USA, [email protected] Lai-Wai Tan Faculty of Civil and Environmental Engineering, University of Tun Hussein Onn Malaysia, Batu Pahat, Malaysia, [email protected] Lei Tang D&P LLC, Phoenix, AZ 85016, USA, [email protected] C. Tenaud LIMSI – UPR 3251 CNRS, BP.133, 91403 ORSAY Cedex, France, [email protected] Dmitry Tereshko Institute of Applied Mathematics, FEB RAS, Vladivostok 690041, Russia, [email protected] Frank Thiele Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, 10623 Berlin, Germany, [email protected] Harvey M. Thompson School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK, [email protected] Svetlana A. Tokareva Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland, [email protected]

Contributors

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Eleuterio F. Toro Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering, University of Trento, I-38100 Trento, Italy, [email protected] S. Townsend School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia, [email protected] Andrey Travin New Technologies and Services, St. Petersburg 195220, Russia, [email protected] Cameron Tropea Chair of Fluid Mechanics and Aerodynamics/Center of Smart Interfaces, Technische Universität Darmstadt, 64287 Darmstadt, Germany, [email protected] Jiyuan Tu RMIT University, Bundoora, Australia, [email protected] E. Turkel School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv, Israel, [email protected] Artur Tyliszczak Czestochowa University of Technology, 42-200 Czestochowa, Poland, [email protected] Arthur Veldman J. Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 9700 AK Groningen, The Netherlands, [email protected] Mula Venkata Ramana Reddy Computational Mechanics Laboratory, JATP, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India, [email protected] Sergii Veremieiev School of Mechanical Engineering, University of Leeds, Leeds LS2 9JT, UK, [email protected] R. Verzicco DIM, Università di Roma Tor Vergata, 00133 Roma, Italy, [email protected] Jan Vierendeels Department of Flow, Heat and Combustion Mechanics, Ghent University, 9000 Ghent, Belgium, [email protected] Paul Vigneaux Unité de Mathématiques Pures et Appliquées, ENS de Lyon, 46, allée d’Italie, 69364 Lyon Cedex 07, France, [email protected] Alexey N. Volkov Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904-4745, USA, [email protected] Jian-Ping Wang State Key Laboratory of Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing, China 100871, [email protected] Tao Wang Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, Canada, [email protected]

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Contributors

Z.J. Wang Department of Aerospace Engineering, CFD Center, Iowa State University, Ames, IA 50011, USA, [email protected] Rik Wemmenhove FORCE Technology Norway AS, 1338 Sandvika, Norway, [email protected] Christelle Wervaecke INRIA Bordeaux Sud-Ouest, Bordeaux, France, [email protected] Marcel Wijnen Faculty of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands, [email protected] Andrew M. Wissink US Army/AFDD, Moffett Field, CA 94035, USA, [email protected] Guoping Xia Purdue University, West Lafayette, IN 47907, USA, [email protected] Zhixiang Xiao School of Aerospace Engineering, Tsinghua University, 100084, Beijing, P.R. China, [email protected] Kun Xu Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China, [email protected] K. Ya. Yakubovsky Scientific and Research Center ECOLEN, Moscow 111116, Russia, [email protected] Nail K. Yamaleev North Carolina A&T State University, Greensboro, NC, USA, [email protected] Makoto Yamamoto Tokyo University of Science, Tokyo, Japan, [email protected] Satoru Yamamoto Department of Computer and Mathematical Sciences, Tohoku University, Sendai 980-8579, Japan, [email protected] Wataru Yamazaki Department of Mechanical Engineering, University of Wyoming, Laramie 82071, USA, [email protected] Yuuki Yamazaki Tokyo University of Science, Tokyo, Japan (currently IHI Cooperation), [email protected] Jaw-Yen Yang Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan, [email protected] S. Yaniv Israel Military Industries Ltd. (IMI), Ramat Hasharon, Israel, [email protected] Takahiro Yasuda The University of Shiga Prefecture, Hikone, Japan, [email protected] Mine Yumusak ROKETSAN Missiles Industries Inc., 06780, Ankara, Turkey, [email protected]

Contributors

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F. Panahi Zadeh Islamic Azad University, Khorramshahr Branch, Khorramshahr, Iran, [email protected] Daniel W. Zaide University of Michigan, Ann Arbor, MI, USA, [email protected] Dmitry K. Zaytsev St. Petersburg State Polytechnic University, St. Petersburg 195251, Russia, [email protected] Leonid V. Zhigilei Materials Science and Engineering Department, University of Virginia, Charlottesville, VA 22904-4745, USA, [email protected] V. Zhukov Keldysh Institute of Applied Mathematics, Moscow, Russia, [email protected] David W. Zingg University of Toronto, Institute for Aerospace Studies, Toronto, ON, Canada M3H 5T6, [email protected] Davide Zuzio Department of Models For Aerodynamics and Energy, ONERA, Toulouse, France, [email protected]

Part I

Plenary Lectures

The Expanding Role of Applications in the Development and Validation of CFD at NASA David M. Schuster

Abstract This chapter focuses on the recent escalation in application of CFD to manned and unmanned flight projects at NASA and the need to often apply these methods to problems for which little or no previous validation data directly applies. The chapter discusses the evolution of NASA’s CFD development from a strict “Develop, Validate, Apply” strategy to sometimes allowing for a “Develop, Apply, Validate” approach. The risks of this approach and some of its unforeseen benefits are discussed and tied to specific operational examples. There are distinct advantages for the CFD developer that is able to operate in this paradigm, and recommendations are provided for those inclined and willing to work in this environment.

1 Introduction Traditionally, Computational Fluid Dynamics (CFD) methods developers have applied a very logical, systematic approach to software development, termed here as the “Develop, Validate, Apply” strategy. In this strategy, “Develop” refers to the coding and verification of the CFD software, while “Validate” refers to the process of running test cases on the software and comparing with known data. These validation data sources can be from other validated CFD methods, sub-scale experiments such as wind tunnel tests, or in rare cases, full-scale data, such as that obtained from flight tests. Upon completion of the validation phase of the strategy, the validated software system moves into the “Apply” stage and is delivered to the end-users who apply the method to problems that fit within the range of validation established for the method. This conservative strategy for CFD development ensures that the users of the method have a fully tested and validated tool for their given application. As long as they don’t stray too far from the parameters under which the method was validated, D.M. Schuster (B) NASA Engineering and Safety Center, NASA Langley Research Center, Hampton, VA 23681, USA e-mail: [email protected]

A. Kuzmin (ed.), Computational Fluid Dynamics 2010, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-17884-9_1,

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they have a tool that is capable of predicting flow characteristics within a demonstrated error band. However, as most anyone who develops CFD methods already knows, end-user applications can quickly migrate away from the parameters under which the software was validated. In some cases, it becomes necessary for end-users to apply the methods to problems that are well outside the validation range of the software. Depending on the criticality of the pending analysis, the users may be forced into a situation where the final stages of the preferred strategy become switched and we find ourselves operating in the less systematic framework of “Develop, Apply, Validate.” This places the end-user in a situation where they have little or no knowledge of neither how the method will operate for their particular problem nor how accurate it will be. As a result, large modeling uncertainties are added to these types of analyses to account for this unknown performance. These additional uncertainties can have a profound impact on the prediction of performance and the overall design of a system. This chapter discusses how one might be required to operate under this strategy and the ramifications of implementing this software maturation strategy.

2 Nasa’s Evolving CFD Maturation Process In the past 7 years, the National Aeronautics and Space Administration (NASA) has seen a marked change in how they apply and ultimately develop and validate their Computational Fluid Dynamics (CFD) methods. This change comes as a result of two major Agency events. The first is the loss of the Space Shuttle Columbia during reentry in 2003. The second is the initiation of NASA’s new exploration vision, embodied in the Agency’s Constellation Program, which requires NASA engineer’s to be responsible for large quantities of the aeroscience data products associated with the program’s crewed spacecraft and launch vehicle. These two events coalesced to fundamentally change how the Agency applies its CFD methods, develops new CFD capability, and validates this capability.

2.1 CFD and the Space Shuttle Program On January 16, 2003 NASA launched the Space Shuttle Columbia on what would turn out to be its 28th and final mission. Roughly 82 s into the flight, a piece of insulating foam from the left bipod ramp on the vehicle’s External Tank (ET) struck the left wing leading edge of Columbia, critically damaging the vehicle. Upon reentry, damage from the debris strike resulted in very high temperature flow reaching the vehicle’s primary structure causing catastrophic structural failure and vehicle breakup over the Southwest United States. Following the accident, a formal investigation was conducted by a board of experts commissioned by the President of the United States. This Columbia Accident Investigation Board (CAIB) sponsored tests and analyses to determine the root cause of the accident and published their findings

Expanding Role of Applications in the Development and Validation of CFD at NASA

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in a report commonly known as the CAIB Report [5]. Among the causes contributing to the accident, NASA’s Organization and Safety Culture were identified as key contributors as “Shuttle Program management made erroneous assumptions about the robustness of a system based on prior success rather than on dependable engineering data and rigorous testing.” In partial response to the accident and the findings of the investigation, NASA established a new organization known as the NASA Engineering and Safety Center (NESC). The NESC operates under the philosophy of three primary tenets to ensure safety: (1) Strong in-line technical checks and balances to ensure proper engineering analysis and data are being applied to the problem. (2) Healthy tension between the Project, Safety, and Engineering components supporting the Program. (3) “Value added” independent assessment of problems that cannot be adequately resolved within the internal Program environment. The NESC is funded through NASA’s Office of the Chief Engineer, which allows it to maintain independence from the programs and projects it is asked to support. The independent assessment functionality of the NESC is one of the unique features of the organization that allows it to provide a technical evaluation of a given situation without biases due to schedule and budgetary pressures. The NESC presently supports 15 discipline-centric teams, each made up of experts from across NASA, other government agencies, industry, and academia. CFD is a key analysis tool of the Aerosciences Technical Discipline Team (TDT) as well as several other TDTs within the NESC. The accident investigation resulting from the Columbia tragedy, as well as the highly increased technical insight into the Space Shuttle Program after it returned to flight status have stretched the Agency’s CFD resources and capabilities. The fast-paced environment of the accident investigation required rapid turn-around of analyses that used our CFD methods on a scale that had not been anticipated during their development. This forced engineers to use the codes in innovative applications, sometimes with limited validation applicable to the specific problems they were analyzing. Since the accident, NASA has successfully flown the Space Shuttle 19 times through May, 2010, see Table 1, and the use of CFD to analyze and evaluate problems on the vehicle has continued to escalate and expand at a very high rate. Examples of CFD usage on the shuttle will be provided for the five bolded flights shown in the table. Far and away, the most dominant use of CFD on the shuttle since the Columbia accident has been for the prediction of launch debris transport. Prior to the Columbia accident, engineers and managers recognized the importance of predicting the transport of foam and ice debris during the ascent phase of the shuttle flight. Tools to predict debris transport were in development prior to the Columbia accident. Early debris prediction techniques used wind tunnel data to provide the aerodynamic environments necessary to make the predictions. In the late 1980s, CFD began to make inroads in predicting these environments [2], albeit on simplified geometry

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D.M. Schuster Table 1 Space Shuttle missions since Columbia accident Mission Launch date Mission Launch date STS-114 STS-121 STS-115 STS-116 STS-117 STS-118 STS-120 STS-122 STS-123 STS-124

7/26/2005 7/4/2006 9/9/2006 12/9/2006 6/8/2007 8/8/2007 10/23/2007 2/7/2008 3/15/2008 5/31/2008

STS-126 STS-119 STS-125 STS-127 STS-128 STS-129 STS-130 STS-131 STS-132

11/14/2008 3/15/2009 5/11/2009 7/15/2009 8/28/2009 11/16/2009 2/8/2010 4/5/2010 5/14/2010

configurations. By the mid-1990s, high-fidelity CFD analyses on much larger, more complex configurations were being used to predict shuttle aerodynamic environments [6, 10]. These simulations matured quickly and generally produced accurate surface pressure predictions when compared with experimental data as shown in Fig. 1 [7]. During this period debris trajectory prediction analyses were performed on a case-by-case basis and a high degree of automation of this process was not available. In addition, the aerodynamic characteristics of the debris itself were not simulated. The investigation of the Columbia accident necessitated the rapid development of an automated debris transport analysis capability. In addition, the CAIB also recommended that the complete debris environment encountered by the Space Shuttle Launch Vehicle during ascent be characterized prior to Return-to-Flight

Fig. 1 High-fidelity Space Shuttle CFD comparison with wind tunnel data (cf. [5])


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(RTF). Since detailed aerodynamic characteristics of the debris components were not included in previous debris transport analyses, these characteristics had to be developed in parallel with the RTF analyses. Various debris shapes were characterized using CFD and these data were validated using ballistic range test results as shown in Fig. 2 [9]. Thus the engineers found themselves validating their methodology virtually in parallel with the analyses being used to prepare the vehicle for

Fig. 2 Comparison of aerodynamic characteristics of Space Shuttle debris computed by CFD with ballistic range data (cf. [6])

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its return to flight. This started to blur the line between apply and validate in the traditional process of CFD method maturation. Once the Space Shuttle returned to operation, this line became increasingly transparent and in some cases, the validate/apply order became completely reversed.

2.2 STS-114 – Space Shuttle Returns to Flight On July 26, 2005, over 2 years after the Columbia accident, NASA launched STS114 marking the Space Shuttle’s return to flight. Given the high sensitivity to debris and the uncertainty in many of the analyses employed leading up to this mission, numerous instruments and cameras were added to the flight manifest. In addition, maneuvers such as the “flip” maneuver of Fig. 3, allowed the vehicle to be photographed and inspected to a level of detail unprecedented in prior flights. As a result of these inspections, a protruding gap filler was discovered between the ceramic tiles on the heatshield near the nose of the spacecraft. From the photographs, dimensions and shape of the protrusion could be derived and used as boundary conditions for the various boundary layer transition and heating analysis tools. These analyses, some involving CFD, were conducted over a span of just a few days while the vehicle was on orbit. The analyses showed the vehicle to be safe for return with the protruding gap filler, but uncertainties in the data produced sufficient concern for the safe return of the vehicle that mission managers formulated and exercised an Extra Vehicular Activity (EVA) to remove the offending component. Astronaut Steven Robinson was attached to the end of the arm of the Space Shuttle Orbiter’s Remote Manipulator System and maneuvered to the front of the vehicle to remove the gap filler, as seen in Fig. 4. This high-risk EVA underscored the importance of accurate analysis capability to future missions and the impact it could

Fig. 3 Space Shuttle performing flip maneuver as it approaches the International Space Station


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Fig. 4 Removal of protruding heatshield gap filler during STS-114 mission

have on the risk posture of these missions. The gap filler analysis was also a harbinger of how our analysis techniques would be stressed and extended during future flights. During STS-114, numerous foam debris events were observed during the vehicle’s ascent indicating that the foam debris issue was not as well understood as presumed prior to the flight. As a result, the program waited nearly a year to conduct its next flight. During this time, several large sources of foam debris were removed from the ET, including two large Protuberance Air Loads (PAL) Ramps. These ramps were part of the original ET design to protect externally mounted cable trays from high velocity crossflows generated during ascent. Considerable testing and analysis, including CFD, were performed prior to STS-114 to demonstrate that the cable trays remained structurally sound without the protection of the PAL ramps. The CFD involved unsteady separated flow over bluff bodies near a ground plane. This complex geometry and flow condition certainly stretched our CFD capability beyond the limits of its validation. Qualification testing to demonstrate that the cable trays could withstand the aerodynamic loads of a space Shuttle launch included unsteady pressure and structural response data that could be used to partially validate the CFD analysis. But again these data were being acquired in parallel with the process being exercised to qualify the system for flight. Thus the validation and application of the CFD were being performed simultaneously. The PAL ramps were identified as a potential debris source prior to STS-114, but the uncertainty of the analysis, testing, and the risk of unintended consequences resulting from removal of the PAL ramps outweighed the risk of a foam strike originating from one of the ramps. During the STS-114 mission though, a large portion of one of the PAL ramps was lost during ascent, Fig. 5. This documented foam loss elevated the risk of the PAL ramps as a foam source, and they were subsequently removed from the ET for the second return to flight mission, STS-121, and subsequently for all future missions.

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Fig. 5 STS-114 PAL Ramp damage sustained during ascent

2.3 STS-121 Returns Debris Transport Prediction Validation Data Due to the large volume of data acquired during STS-114 and the resulting analyses, testing, and system changes precipitated by this flight, STS-121 was launched nearly a year after STS-114. Imagery data acquired during STS-121 demonstrated that the number of foam liberation events had been significantly reduced and the mission proceeded with few flight issues. The relatively clean flight provided engineers with an opportunity to perform more detailed analyses of some of the data that had been collected on both STS-114 and STS-121. In particular, some of the debris transport methodology that had undergone such rapid development could finally be compared against flight data [3]. Figure 6 shows two images of foam debris captured during the STS-121 launch that was liberated from one of the Space Shuttle ET ice/frost ramps. The debris travels aft and passes near the starboard Solid Rocket Booster (SRB) nozzle. The video imagery, in this case from a camera mounted near the nose of the SRB, is of sufficient quality and resolution to roughly estimate the size and trajectory of the foam debris. Figure 7 shows a predicted trajectory for a piece of foam released from the same ice/frost ramp at this point in the ascent trajectory with the debris ultimately passing right over the top of the SRB as observed in flight.


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Fig. 6 STS-121 ascent imagery of foam debris liberated from the Space Shuttle ET (cf. [7])

Predicted Debris Path

Fig. 7 Predicted debris path for foam debris liberated from a specific ET ice frost ramp (cf. [7]). Reprinted with permission of The Aerospace Corporation

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The ability to correlate computational methods against full-scale flight data in this manner is often very difficult to accomplish in the standard “Develop, Validate, Apply” maturation strategy due to the cost of conducting a flight test solely for the purpose of method validation. However, when the application and validation tasks are performed in a more parallel fashion as with the foam debris trajectory prediction, full-scale validation opportunities are often more forthcoming, particularly if the method developers are tightly integrated into the application and the definition of the flight measurements that could be used for validation.

2.4 STS-118 – Tile Damage Due to Debris During the post launch inspection of STS-118, significant heatshield tile damage was observed on a pair of tiles on the starboard wing as shown in Fig. 8. Digital photographs taken from aboard the International Space Station (ISS) as the shuttle approached and performed its flip maneuver showed that the debris had dug into the

Fig. 8 Tile damage incurred during the launch of STS-118


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tile enough to expose the underlying felt blanket that is used to mount the tiles to the vehicle’s metallic structure. Thus, a small section of the vehicle was left without tile protection for atmospheric reentry. Measurements and geometry for the divot were estimated from the photographs and mathematical and physical models of the damage were constructed. CFD was used to predict heating for the damaged area during entry, a sample of which is shown in Fig. 9. The methods used to make these predictions had been validated for a variety of divot shapes, but this divot had a particularly deep, complex, two-tiered geometry, putting the heating tool predictions outside their validation range. The heating tools predicted that the vehicle could reenter the atmosphere safely with the damage, but the uncertainty in the analysis required that further testing be performed in NASA’s arc-jet test facilities before the vehicle could be cleared for entry. A model of the damaged tile system, shown in Fig. 10, was constructed and quickly tested. Data from the heating tool were used to set the arc-jet flow conditions with significant margin beyond the heating levels predicted by the CFD. The conservative test conditions showed that the surrounding heatshield tiles would be damaged, but would not lose structural or thermal integrity. In addition, the exposed portion of the underlying metallic structure would not see temperatures high enough to damage the vehicles primary structure. The CFD analysis coupled with the conservative test data provided engineers and managers with the confidence to allow the vehicle to reenter the atmosphere without the need for a high-risk, high-uncertainty repair. Ultimately the Space Shuttle Orbiter successfully returned to earth with little indication of damage due to excessive heating in this area.

Fig. 9 CFD prediction of heating due to STS-118 tile damage

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Fig. 10 Model of STS-118 tile damage for use in arc-jet testing

All geometry modeling, model construction, analysis and testing for this effort were conducted in a matter of days while the vehicle was on orbit. Due to the complexity of the damage geometry, the heating tool used to predict the reentry temperatures at the damage site was used outside its validated range of applicability. But even though engineers and managers felt it necessary and prudent to conduct testing at conservative conditions to ensure the integrity of the thermal protection system, the CFD analysis played a significant role in the overall decision process to reenter the vehicle without repair. The method developers received some validation data on this geometry from the arc-jet testing prior to reentry, but in this case the application of the CFD had clearly begun to edge ahead of the method validation. Engineers and managers realized the value of the time critical data that could be extracted from the methodology even if it was applied beyond its validated range.

2.5 STS-124 – Launch Pad 39A Flame Trench Damage The launch of STS-124 caused significant damage to Kennedy Space Center’s Launch Pad 39A. Shortly after SRB ignition, 3,500 fire bricks were torn from the east wall of the Pad 39A SRB flame trench and exhausted north away from the launch pad, see Fig. 11. Each of these bricks weighed approximately 20 lbm (9 kg) and brick fragments were clocked using radar at up to 1,000 ft/s as they exited the flame trench. Beyond the damage to pad facilities, such as the flame trench, fencing, and other ground equipment, there was concern whether brick fragments, if liberated on future flights, could somehow make their way back to the Space Shuttle and damage it during its liftoff from the pad. CFD analysis was used to provide two critical sets of data in the subsequent investigation and for the flame trench repair. First, it was used to predict the


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Fig. 11 Kennedy Space Center Launch Pad 39A damage after launch of STS-124

time-dependent pressure and temperature environment along the walls of the flame trench as depicted in Fig. 12. Second, it was used to predict the flowfield properties within the trench for use in debris transport analysis, as seen in Fig. 13. This problem posed some extreme challenges for the CFD methods and those trying to apply them. First, the geometry is very complex. The SRBs fire through a pair of holes in a part of the pad known as the Mobile Launch Platform (MLP). The MLP and the lower portion of the Space Shuttle SRBs are shown as the gray geometry in Fig. 12. The two SRB plumes pass through the holes in the MLP and impact on the main exhaust flame deflector as depicted in Fig. 13. The main flame deflector redirects the plume horizontally into the flame trench and allows it to exhaust out the north side of the pad and away from the launching vehicle. The other side of the main exhaust

Fig. 12 CFD prediction of the flow environment along the walls and floor of the Pad 39A flame trench shortly after SRB ignition

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Fig. 13 CFD prediction of SRB plume and associated flowfield during launch of STS-124

deflector redirects the three Space Shuttle Main Engine (SSME) plumes to the south and away from the vehicle. In addition to the geometric complexity, there is significant flow physics complexity as well. The SRB plumes involve chemically reacting species exhausted at very high speeds and interacting with still air. The SRBs generate a severe ignition overpressure transient at startup that makes its way down the flame trench and is believed to have initiated the separation of the bricks from the east wall of the flame trench. Finally, the launch process involves the injection of an extremely large volume of water into the SRB holes and across the top surface of the MLP deck to provide acoustic suppression during the launch. These physical and geometric features posed a severe challenge to the application of CFD, and forced its use well beyond the limits of validation and previous experience. The analysis was performed in a time accurate mode with two gases, one for the SRB plumes and the other for the surrounding air. Neither chemical reactions in the plumes nor any attempt at simulating the water deluge were included in the analysis. The initial pressure and temperature transients predicted along the flame trench wall were used to provide structural design data used in the subsequent flame trench repair. The steady state flow data was used to feed debris transport predictions should a similar event occur in the future. The design for the flame trench repair was based on the CFD loads with significant margin included for modeling uncertainty. Similarly, the debris transport analysis involved significant deviations from the nominal predicted environments in the trench to ensure that a conservative estimate of the probability of debris


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striking the vehicle could be obtained. Again, this analysis and design was done on a very compressed schedule as repairs had to be accomplished prior to the next shuttle mission and the vehicle had to be cleared for debris impact before it could be launched. In the end, a repair could be formulated and applied to the launch pad on a schedule that did not impact the next launch and the debris transport analysis showed no credible mechanism for debris to impact the vehicle. The significant point here is that the CFD analysis was knowingly applied well beyond its known range of applicability, and this was compensated for in the final design and debris analysis by adding large uncertainty margins to the predicted data. This approach had an unexpected beneficial aspect. The high uncertainties in the predictions provided motivation to acquire data for comparison with the flow predictions to help validate them and the designs and analyses they influenced. Instrumentation was added to the flame trench to measure key environments during the subsequent launch. Greater focus was placed on obtaining and analyzing debris transport information near the pad during subsequent launches as well. Thus, when engineers took the risk of applying their methods outside their comfortable range of validation and generated results with acknowledged high levels of uncertainty, they found that they could more effectively influence the acquisition of needed validation data for their application.

2.6 STS-126 – Flow Control Valve Poppet Failure The final Space Shuttle example discussed in this chapter is the failure of an Orbiter internal component that is used to control the pressure in the Liquid Hydrogen (LH2) propellant tank during launch. The LH2 tank, located in the ET, is pressurized using a Gaseous Hydrogen (GH2) bleed from the SSMEs. Three Flow Control Valves (FCV) control the flow of GH2 back into the LH2 tank and thus manage the pressure in the tank. During the launch of STS-126, one of the FCVs transitioned to a high flow state without being commanded to do so and remained in that state for the remainder of the flight. Since the FCVs are located in the Space Shuttle Orbiter, they could be removed and inspected after the mission. This inspection revealed that a piece of the poppet on the suspect FCV had broken loose during the launch, as shown in Fig. 14. The potential adverse consequences of this type of damage are twofold. First, with the broken poppet, the given FCV is free to pressurize the LH2 tank continuously and if the break is large enough or multiple FCV poppets simultaneously fail, the LH2 tank could be over-pressurized with catastrophic consequences. Second, if the energy and size of the poppet fragment is large enough, the fragment could be propelled downstream in the pressurization system with enough momentum to potentially puncture the GH2 repressurization piping as it tries to negotiate the various turns, bends, and diameter changes along the way from the Orbiter to the ET. This piping runs through areas in the orbiter, which if subjected to a gaseous hydrogen leak, could result in external combustion of the hydrogen, again with

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Fig. 14 Inspection of SSME FCV after STS-124 mission reveals broken poppet

catastrophic results. In fact, borescope inspection of the pipes in the Orbiter downstream of the FCV showed several dents and scratches that were concluded to be caused by the failed poppet fragment as it made its way through the pipes. The fragment itself was never found and it was assumed to have been transmitted all the way to the ET where it was lost after stage separation. Figure 15 shows a cutaway schematic of a FCV with the GH2 flow path indicated. GH2 enters the system from the top at very high pressure and flows around the poppet, which is translated left and right in the figure to open and close the flow path. The GH2 then enters the outlet tube and flows though a long system of pipes to the LH2 tank in the ET. The poppet position is controlled by a solenoid in the FCV. When the subject poppet fragment broke off, it created an open flow path for GH2 that could not be shut off. CFD and debris transport analysis were again key contributors to the investigation of this problem and the risk analysis for future flights that might encounter a similar issue. The FCV flow path, including the flow through much of the downstream piping was modeled using CFD. Both nominal and broken poppet configurations were analyzed and steady state and transient analyses were performed. Figure 16 shows a two-dimensional steady state CFD analysis of the flow through a nominal FCV with an intact poppet. The stream traces show the complex nature of the predicted flowfield downstream of the poppet. There is a very high pressure ratio between the flow upstream of the poppet and that downstream resulting in a very strong jet that emerges from the poppet and impacts the wall of the outlet tube. Two regions of recirculating flow are generated by this jet, a clearly visible, large region below the jet and directly behind the poppet and a smaller area in the triangular region above the jet. Figure 17 shows a transient analysis of the developing flowfield behind a broken poppet just after it has fragmented. In this case, the poppet was fully closed and there was low flow mass through the poppet when it broke. From this analysis, classical one-dimensional flow features such as the propagating shock and contact


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FLOW

OUTLET FLOW Poppet Stroke = 0.017″

Fig. 15 FCV schematic showing GH2 flow path and poppet control valve

Fig. 16 Two-dimensional steady CFD analysis of the nominal FCV operation

discontinuity can be identified in the more complex two-dimensional flow. This type of information was used to estimate the downstream acceleration of the poppet fragment immediately after it was released from the poppet. These data were then input to a modified version of the debris transport methodology to predict the trajectory and velocity of the fragment as it moved downstream. This analysis required that not only the CFD be stretched well beyond its validated range of application, but also the debris transport methodology used to predict the trajectory of the poppet. In fact, new debris transport capability for pipe flows with multiple impacts was developed on the fly as the analysis progressed. Some testing was conducted in parallel with the debris trajectory application and development, but this testing was very cursory in nature. The debris simulations were used to perform a Probability Risk Assessment (PRA) for poppet debris released from a FCV. From the PRA, guidelines were established for the maximum size of poppet debris that could be tolerated without sustaining collateral damage to the

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Fig. 17 Transient CFD analysis of FCV flow just after poppet has broken

downstream pipes. Ultimately the problem was solved by requiring highly detailed inspection of the poppets prior to each flight, paying particular attention to poppets that have been used a high number of times.

2.7 Space Shuttle Lessons Learned During the Columbia accident investigation, engineers found that the CFD codes provided data that appeared to be physically realistic, but they often only had limited test data to quantitatively verify their answers. Ultimately, many of the analyses performed during the accident investigation showed sufficient promise that significant efforts were undertaken to improve tools in anticipation of broader support of future flight operations. This provided the developers of these tools an opportunity to take a step back and formulate experiments and strategies to further and more rigorously validate and calibrate the new tools. However, the increased technical insight into post-Columbia Space Shuttle flights only generated a broader range and greater diversity of problems that required innovative use of our CFD methods. In addition, these applications were all performed on a time-critical basis since they required answers to be generated while the vehicle was on orbit, or in the days and weeks leading up to a launch. NASA engineers sometimes found themselves employing their CFD methods to previously unanticipated applications with minimal quantitative data against which to measure their results. If a given on-orbit or preflight problem generated a longterm requirement for a future similar capability, then testing could be formulated and conducted after the fact to help validate the new capability and bring it into the stable of tools used to support future flights. If the problem was deemed to be a one-of-a-kind application with little or no requirement for future application, then no data to support the application might ever be generated. Regardless of its future applicability, the CFD analyses of these time-critical problems were often the only data that could be generated on a time-scale necessary to address the problem


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and with sufficient accuracy to characterize the issue. Thus NASA engineers and researchers sometimes found themselves shifting into the “Develop, Apply, Validate” paradigm. From a technical rigor standpoint this is a less than ideal approach to developing new capability, but interestingly, the approach has opened doors for developers to acquire data that generally is extremely difficult to obtain in the more ideal “Develop, Validate, Apply” maturation strategy; namely full-scale operational and flight data. With the CFD techniques being applied in this increasingly aggressive fashion, engineers were forced to add generous uncertainty margins to their results to account for the fact that they didn’t have sufficient data or experience against which to benchmark their analyses. It became commonplace to see 20–30% additional margin added to computational results as a “modeling uncertainty factor.” Obviously adding this type of margin has operational and vehicle performance implications. As a result, Project Managers often become more motivated to acquire data to reduce these uncertainty margins. Often this data comes from the flight system it most directly impacts through additional instruments and data systems on the flight vehicle. Flight data has always been the gold standard for validation of CFD methods, but acquisition of this type of data is typically difficult, if not impossible, to obtain in a focused, standalone CFD development effort. Thus the aggressive application of NASA’s CFD methods within the Space Shuttle Program has led to an increased number of opportunities to obtain highly sought after flight data for method evaluation and validation; a turn of events that certainly could not have been envisioned a priori.

2.8 NASA’s Constellation Program Further Reinforces the Paradigm Shift This evolutionary process has been continued with NASA’s efforts to provide a broad array of aerodynamic and aerothermodynamic data products in support of the Agency’s Constellation Program (CxP) manned spaceflight initiative. CxP designs have pressed engineers to predict the performance of aerodynamically complex vehicles, again with minimal applicable validation data against which to benchmark their methods. The CxP vehicles require analysis of a broad spectrum of flow issues, ranging from highly separated, bluff body flows to multiple plume jet interaction flows. Speed ranges run the gamut from the very low velocities of launch pad winds through transonic flight during vehicle ascent to hypersonic reentering flight. The CxP has focused its early efforts on the development of two primary components, the Orion spacecraft and the Ares I launch vehicle. The Ares I, depicted in Fig. 18, consists of a Space Shuttle heritage solid rocket booster as a first stage and a larger diameter liquid rocket second stage. From a CFD standpoint, major challenges posed by the Ares I include unsteady separated flows generated by the diameter change in the first/second stage transition and protuberances located at numerous locations on the vehicle. These unsteady separated flows are particularly

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Fig. 18 Artist concept of the Ares I launch vehicle

important for definition of aeroacoustic and buffet environments for structural loads. There are also a number of smaller reaction control motors incorporated on the vehicle that perform tasks like vehicle roll control and stage separation. Ground wind loads are also important to the vehicle as it rolls out and sits on the launch pad prior to flight. Since the SRB first stage is designed to be recovered and refurbished for flight, similar to the Space Shuttle, it must tumble after stage separation. Otherwise, if it were to trim nose or tail first, the recovery parachutes would likely fail due to high opening loads. Therefore, by definition, the tumbling flight of the separated first stage involves significant unsteady separated flow. Finally, the Ares I is a very high length to diameter ratio vehicle. This presents significant modeling problems given the small details, such as protuberances, that are important to the vehicle performance and must be modeled with sufficient resolution to capture their induced aerodynamics. As a result, the Ares I grid models are extremely large and costly to analyze.


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Fig. 19 NASA’s Orion spacecraft

By outward appearance, the Orion spacecraft, shown in Fig. 19, is similar in shape to NASA’s Apollo spacecraft, but it is much larger and heavier. Aerothermodynamic issues during reentry dominate the CFD analysis of this vehicle. At present, the majority of design effort is being spent on earth entry from Low Earth Orbit (LEO), as in a mission to the ISS. However, the vehicle is also expected to operate beyond LEO, where earth entry will be at much higher enthalpy. This brings issues for which CFD is not highly developed into the analysis space, such as radiation effects. Reaction Control System (RCS) jet interaction with the surrounding vehicle aerodynamics is also important for this vehicle. CFD is not well developed for the hypersonic separated flow jet interactions posed by this problem. In addition to the basic aerodynamics of the Orion spacecraft, designers must also analyze the performance of the Orion Launch Abort Vehicle (LAV), shown in Fig. 20. The LAV is designed to provide an emergency crew escape capability in the event of a launch vehicle failure. It must be able to be operated anywhere in the flight trajectory from sitting stationary on the launch pad until shortly after first stage burnout. The vehicle is statically unstable and requires an active flight control system to maintain vehicle orientation throughout its flight. The LAV is powered by a main Abort Motor (AM) exhausting from the four large nozzles about midway up the forward tower. At the forward end of vehicle is the Attitude Control Motor (ACM), which is a single solid rocket motor that exhausts through eight nozzles arranged circumferentially around the top of the tower. Each of these eight nozzles can be metered individually to actively control the pitch and yaw attitude of the LAV. This control system and arrangement of nozzles has proven to be exceptionally difficult to analyze and predict vehicle performance. When both the AM and ACM

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Fig. 20 Orion Launch Abort Vehicle (LAV)

are operating together, the ACM plumes can interact with the AM plumes as shown in Fig. 21. The case depicted by the figure shows all eight ACM plumes firing simultaneously, but in reality, the ACM nozzles can be metered so that individual or select groups of nozzles can be operating while the others are effectively shut off. In these cases, the ACM jets can push on the AM plumes that then push on the back of the LAV to generate pitching or yawing moments on the vehicle in addition to simply the reactive moments due to the thrust of the individual ACM nozzles. These interactions are highly nonlinear and dependent on the flight conditions, and are particularly severe in the transonic flight regime. The application of CFD to this flow is problematic due to the lack of experience and data for these types of interactions. The problem is compounded by the fact that wind tunnel testing of this type of interaction is very difficult, complicated, and expensive and in the end, it is virtually impossible to match flight conditions for this type of problem in a subscale wind tunnel test. CFD is the engineer’s best tool to accurately predict these flows under full-scale flight conditions, but with the lack of data and experience in this problem, the uncertainties surrounding the CFD analyses are large. An example of how this issue is challenging the Orion LAV developers is presented in Fig. 22. This figure shows the CFD predicted flowfield for the Orion LAV with the main abort motors running and two of the ACM nozzles firing downward. At certain conditions in the transonic flight regime, the two downward-firing ACM jets become left-right asymmetric, even though geometrically and mathematically they should remain symmetric. In this figure, the top two images show the forward tip of the LAV tower with a slice through the flow just aft of the ACM nozzles. The flow slice shows Mach number contour in the plane, and the shadowed areas are a


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Fig. 21 Orion LAV AM/ACM interaction

species iso-surface showing the shape of the two ACM plumes. The asymmetry in the plume development is evident in these two images. This predicted asymmetry greatly reduces the pitch effectiveness of the ACM and produces a yawing moment on the vehicle when a pure pitch is intended from this nozzle firing combination. Depending on how the CFD method is initiated and executed, the asymmetry can be generated to the right or left, and in some cases the flow can remain symmetric. Code-to-code comparisons show that this asymmetry is predicted consistently by all the CFD methods applied to the problem to date. Engineers are struggling with how to account for this phenomenon in the vehicle design. There are questions whether the phenomenon will ever surface during normal operations given the constantly changing flow conditions and ACM operating parameters. Even if it does occur during flight, the question becomes whether it will persist long enough to adversely affect the LAV performance. Issues like this are having a detrimental impact on the vehicle designs as large performance margins must be built into the concepts with consequential penalties in weight and system complexity. Unlike the design and development of NASA’s last manned spacecraft, the Space Shuttle, CxP is relying heavily on CFD to make design decisions and predict vehicle performance. The program is performing considerably less testing than during the Apollo and Space Shuttle programs and as a result we see engineers again applying large modeling uncertainty factors on analysis data that impacts vehicle designs. Testing that is being performed is serving the dual

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Fig. 22 Orion LAV flow asymmetry predicted by CFD

role of vehicle performance evaluation and method validation, but there simply isn’t enough testing available to adequately validate and calibrate the wide range of CFD applications required for this program. In the case of the aforementioned ACM asymmetry, it is not clear that the phenomenon could be accurately predicted or characterized in the wind tunnel. So again, NASA is highly motivated to use the flight tests they have scheduled in support of the program to also acquire data that can be used to reduce the modeling uncertainty carried in the development of the vehicle. In the last year, CxP has flown three test flights, the Max Launch Abort System, the Ares I-X demonstration flight and the Pad Abort 1 (PA-1) launch pad abort flight. Given the broad use of CFD and other analytical methods and the large modeling uncertainties associated with these analyses, each of these vehicles have been liberally instrumented with aerodynamic and aerothermodynamic sensors to capture steady pressure data, buffet pressure data, aeroacoustic pressures, and heating data, as well as flight telemetry information from which flight conditions and integrated force and moment data can be derived. So NASA CFD application engineers and development researchers suddenly have new sources of flight data against which to evaluate their capability and methods. Flight data can be extremely difficult and rare to acquire in the normal “Develop, Validate, Apply” CFD maturation cycle. But through the liberal application of our CFD capability, particularly for cases


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where little or no validation data or past experience is available, developers can suddenly find themselves in a situation where managers are highly motivated to provide ground test and flight data that can be used to help validate their methods and reduce uncertainties. In some areas, engineers and researchers have begun to identify Flights-ofOpportunity where existing flight projects are evaluated and targeted for their potential to produce needed flight data for methodology validation. As a result of this type of forward thinking, boundary layer transition experiments have recently been performed on the Space Shuttle [1, 8] as it reenters the atmosphere. Instrumentation has also been added to the flight manifest of the Mars Science Laboratory entry vehicle to measure heatshield performance parameters as it enter the Martian atmosphere [4].

3 Requirements, Pros, and Cons of the “Develop, Apply, Validate” CFD Maturation Strategy When the validate and apply processes begin to merge and switch, engineers and researchers who have been classically known as CFD developers must begin to work more closely with those applying the CFD methods. In the classical paradigm, a developer took his methodology all the way through the validation stage and when complete, delivered an analysis capability that had been tested for a specific range of problems. Those applying the methods knew this range of applicability and had confidence, and proof, of the method’s ability to predict these specific problem classes. When the codes begin to be applied outside their known range of validation, engineers look to the people most experienced with the method, the developers, to help guide them through the application. Suddenly developers find themselves answering difficult questions about specific problems pertaining to specific vehicle designs. As a result of this closer interaction, the developer finds themselves performing tasks differently than in the past. First, they find themselves analyzing and understanding more cases than the handful of validation cases tested under the classical strategy. Vehicle applications typically involve parametric analyses of hundreds if not thousands of cases, a handful or range of which might generate some poorly understood flow phenomenon or unanticipated performance. The application engineers thus look to the developers to help them determine if the methodology is indeed capable of predicting these events and if they are physically realistic or an artifice of the numerical methodology. This forces developers to exercise cases on geometries that are often much more complex than the typical validation case. To sufficiently refine these geometries, the grids are also usually much larger than for the typical validation cases analyzed. For instance, it is not unusual for grids on the Ares I or Orion LAV to exceed 50 million grid points. Analyzing data on these more complicated geometries and larger grids put a greater burden on post processing and interpretation of the data. The physical

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characteristics of typical CFD validation cases are often very well known by the development community and it is usually apparent whether a given methodology is behaving reasonably. This is not the case when the applications begin to outpace the validation. Physical characteristics of these applications are usually not very well understood and it often takes a large amount of post processing to cast results in a form that best highlights these characteristics. Finally, as the physical characteristics and the performance of the methodology become better understood, opportunities for ground and flight testing begin to emerge. These tests are typically focused on predicting the performance and reducing uncertainty of a vehicle or system, but when analysis uncertainties are high, they also become opportunities to validate the methods used to predict performance. Again, the application engineers look to the developers to help them define and place instrumentation so as to best capture the physical events that are challenging the methods. As a result, the developer can also find themselves becoming more involved in the tests of the project, as opposed to tests formulated specifically to validate methodology. In the “Develop, Apply, Validate” strategy, the developer must become more aligned with the application engineers and their problems. There are definite pros and cons to this alignment. Many of the pros have been highlighted previously in this chapter, the most prevalent of which is that the developers become better aware of the problems their methods are being asked to analyze and they receive direct input as to the problems that require address for future development. Access to validation tests can also be improved. Acquiring sufficient test data is always a struggle under the classical “Develop, Validate, Apply” strategy; acquiring flight data under this paradigm is very rare. Also, the methods get to the applications engineers faster because they don’t go through the validation process before they begin application. Many of the cons are fairly obvious. In the extreme, methods could be introduced to the engineering community before they have undergone even the most basic of validation. At best, applying methods outside their known validation range is higher risk and the associated large uncertainties associated with this strategy are warranted. The full development cycle for methodology is slower using this strategy since significant application of the method may have occurred prior to the acquisition of sufficient validation data to certify the methodology. The development cycle can also become a more organic process with it becoming increasingly difficult to define the beginning or the end of the cycle. This could be seen as either a pro or a con, especially when one considers the effort required in advocating and initiating a new development campaign; a self-sustaining process can be very attractive.

4 Conclusion NASA is experiencing an explosive growth in CFD applications that is outpacing the validation of their methods. Two large reasons for this growth are the increased technical insight into Space Shuttle operations as precipitated by the Columbia


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accident and the development of new manned spacecraft under the Constellation Program. Under these programs engineers are applying CFD to problems that were not or could not be foreseen in the standard development/validation cycle. The highpriority, time-critical nature of these analyses do not provide sufficient opportunity to validate the methods prior to their application. Application of the methods in these environments lead to large uncertainties and design margins being applied to the vehicles and systems to offset the lack of knowledge and experience in the specific application of the CFD methodology. These margins can have a significant impact on vehicle performance and reducing uncertainty becomes a critical issue to vehicle operations and designs. To reduce uncertainties, ground and flight testing is formulated and CFD developers suddenly find themselves in a position to acquire the much-needed data to validate their methods. Thus while there are definite costs to operating in a “Develop, Apply, Validate” strategy, there are ancillary benefits as well. NASA has naturally evolved into this strategy as a result of mission events and mission requirements. It is unlikely that anyone would have adopted this approach as an a priori strategy, simply because it does not fit the well-established logical and systematic approach to method development. But now that the Agency has refined and exercised this strategy for several years and started to realize some of its benefits, it warrants a closer look as an alternative approach to method development. It can be particularly attractive in environments where advocacy for tool development can be a challenge.

References 1. Anderson, B.A., et al.: Boundary Layer Transition Flight Experiment Overview and In-Situ Measurements. AIAA Paper 2010-0420, Jan 2010 2. Buning, P.G., et al.: Numerical Simulation of the Integrated Space Shuttle Vehicle in Ascent. AIAA Paper 88-4359-CP, Aug 1988 3. Eby, M.A.: Aerodynamics Module for the Space Shuttle Foam Debris Probabilistic Risk Assessment. AIAA Paper 2008-6909, Aug 2008 4. Gazarik, M.J., et al.: Overview of the MEDLI Project. IEEEAC Paper #510, Mar 2008 5. Gehman, Jr., H.W., et al.: Columbia Accident Investigation Board, Report vol. 1. National Aeronautics and Space Administration, Aug 2003 6. Gomez, R.J., Ma, E.C.: Validation of a Large Scale Chimera Grid System for the Space Shuttle Launch Vehicle. AIAA Paper 94-1859, June 1994 7. Gomez, R.J., et al.: STS-107 Ascent CFD Support. AIAA Paper 2004–2226, July 2004 8. Horvath, T.J., et al.: The HYTHIRM Project: Flight Thermography of the Space Shuttle during Hypersonic Re-entry. AIAA Paper 2010-0241, Jan 2010 9. Murman, S.M., Aftosmis, M.J., Rogers, S.E.: Characterization of Space Shuttle Ascent Debris Aerodynamics Using CFD Methods. AIAA Paper 2005-1223, Jan 2005 10. Pearce, D.G., et al.: Development of a Large Scale Chimera Grid System for the Space Shuttle Launch Vehicle. AIAA Paper 93-0533, Jan 1993

Thermodynamically Consistent Systems of Hyperbolic Equations S.K. Godunov

Abstract A discussion of the author’s advanced viewpoint on the underlying principles for construction of Godunov’s scheme is presented. The application of these principles in problems of elastic and elastoplastic deformations is outlined. The presentation is based on extensive numerical simulations performed for both an analysis of the solution convergence details with decreasing mesh step size (for equations of Fluid Dynamics) and the motivation of modeling an elastoplastic media by an “effective elastic deformation” and the employment of Maxwell’s viscosities. Being engaged for 58 years in gas dynamics problems and in more general ones of continuum mechanics, also developing algorithms for numerical simulation of unsteady phenomena, I understood the need to directly construct discrete models for a media composed of elementary microcells rather than to rely only on commonly used differential equations available in monographs and textbooks. In this way, one should aim at obtaining such a behavior of the macro objects (composed of the microcells) that would realistically enough resemble the phenomena in the actual media to be approximated by our model. In fact, exactly this kind of modeling had been performed in “Godunov’s scheme” in 1953–1954 (published in 1959), though the above interpretation of the scheme was elicited some later. In the current lecture, I would like to deliver my up-to-date understanding of the scheme and describe its extension to problems of elastoplastic deformations, which has been developed together with my colleagues. This lecture can be viewed as a sequel of the one given at Michigan University in May 1997 and published in Novosibirsk [5]. A condensed version of the paper [5] came out in English in J. Comp. Phys. [6], whereas a translation of the whole paper into English was published by INRIA in 2008 [7]. The overdetermination of equations governing smooth solutions turned out to be of crucial importance. At the same time, for modeling discontinuities of solutions

S.K. Godunov (B) Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, 630090 Novosibirsk, Russia e-mail: [email protected]


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(shock waves), one needs to replace an equation by an inequality (the entropy rise), whereas for the smooth solutions the overdetermination remains valid. The lecture is focused on the points as follows: 1. The structure of fluid dynamics equations as well as a number of other systems of mathematical physics, which are noticeably diversified. These equations are typically overdetermined with respect to the smooth solutions, whereas the overdetermination used to disappear with respect to the discontinuous ones [1, 3, 4]. In the second case, an equation (the entropy conservation law) turns into an inequality, as the discontinuity relations should often take into consideration dissipative phenomena inside the microscopic zone being modeled by the discontinuity. 2. The symmetric hyperbolicity of the systems at hand (for the smooth portions of solutions). This hyperbolicity is usually a corollary of thermodynamics laws [2, 9, 10]. 3. A detailed discussion of the original Godunov’s scheme structure on the basis of the ideas introduced in items 1 and 2. Feasible simplifications, e.g., a linearization of the used Riemann problem, implicit versions of the scheme, and costs to be paid for these modernizations [11]. 4. Thermodynamically consistent modeling the equations of elasticity and the ones of elastoplastic deformations using a Maxwell scheme; the modeling has been developed up to a difference realization based on the same principles as those in gasdynamics [8, 10]. 5. A review of conclusions inferred from the analysis of the solution convergence with mesh refinement; a discussion of the convergence concept itself. The weak convergence (with mesh refinement) of approximate solutions of the gas dynamics equations modeled with the standard Godunov’s scheme has been studied. Though this scheme is of the first order in the step size h for smooth solutions, numerical simulations have shown that, when modeling discontinuous solutions, the order of accuracy can be estimated by h 0.8 for principal conservation laws, and by h 0.5 for the entropy. It turned out that this accuracy is attained in both cases: when one considers exact solutions of the Riemann problem, and when the problem is solved approximately. We refer the reader to the paper [11] for a detailed description of the above mentioned numerical simulations. Acknowledgements This work was supported by a grant no. HIII-9019.2006.1 under a program “Leading Scientific Schools” of the President of Russian Federation, and by an interdisciplinary contract of the Presidium of the Siberian Branch, the Russian Academy of Sciences, Project no. 40.

References 1. Godunov, S.K.: On the concept of generalized solution. Soviet. Math. Dokl. 1, 1194–1196 (1960) (Trans: Dokl. Akad. Nauk SSSR 134, 1279–1282, 1960) 2. Godunov S.K.: An interesting class of quasilinear systems. Soviet Math. Dokl., 2, 947–949 (1961) (translation from: Dokl. Akad. Nauk SSSR 139, 521–523, 1961).

Thermodynamically Consistent Systems of Hyperbolic Equations

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3. Godunov, S.K.: On nonunique “blurring” of discontinuities in solutions of quasi-linear systems. Soviet Math. Dokl. 2, 43–44 (1961) (Trans: Dokl. Akad. Nauk SSSR 136, 272–273, 1961) 4. Godunov, S.K.: The problem of a generalized solution in the theory of quasilinear equations and in Gas Dynamics. Russ. Math. Surv. 17(3) (Trans: Uspekhi Matematicheskikh Nauk 3(105), 147–158, 1962) 5. Godunov, S.K.: Reminiscences About Numerical Schemes, p. 29. Nauchnaya Kniga, Novosibirsk (in Russian) (1997) 6. Godunov, S.K.: Reminiscences about difference schemes. J. Comput. Phys. 153, 6–25 (1999) 7. Godunov, S.K.: Reminiscences About Numerical Schemes, p. 25. Rapport de recherché No 6666: Theme NUM. Institut National de recherché en informatique et en automatique. Le Chesnay Cedex (France) (2008) 8. Godunov, S.K., Peshkov, I.M.: Symmetric hyperbolic equations in the nonlinear elasticity theory. Comput. Math. Math. Phys. 48, 975–995 (2008) 9. Godunov, S.K., Peshkov, I.M.: Symmetrization of the nonlinear system of gasdynamics equations. Sib. Math. J. 49(5), 829–834 (2008) 10. Godunov, S.K., Peshkov, I.M.: Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium. Comput. Maths. Math. Phys. 50, 1409–1426 (2010) 11. Godunov, S.K., Manuzina, Yu.D., Nazar’eva, M.A.: Experimental analysis of convergence of the numerical solution to a generalized solution in fluid dynamics. Comput. Math. Math. Phys. 51(1), 88–95 (2011)

Part II

Keynote Lectures

A Brief History of Shock-Fitting Manuel D. Salas

Abstract The development of shock-fitting techniques for computational fluid dynamics over the last 50 years is reviewed.

1 Early Attempts at Computing Flows with Shocks Applied mathematics changed forever with the onset of the Second World War. Prior to the war, the main business of mathematicians and physicists was the formulation of the partial differential equations governing some physical phenomenon. In fluid mechanics, this was accomplished by men like d’Alembert, Euler, Navier and Stokes in the eighteenth and nineteenth centuries. To obtain meaningful solutions from these equations, it was necessary to formulate the thermodynamic laws obeyed by fluids and gases. The foundation for this was laid in the later part of the nineteenth century. In the early twentieth century, problems in fluid mechanics were solved by finding solutions in-the-large, i.e., looking for functions that satisfied the partial differential equation for somewhat general initial and boundary conditions. Knowledge of these formal solutions provided a global understanding of the structure of the solution. The dirty business of finding solutions to specific problems was usually left to engineers. However, the problems of interest around the time of the Second World War were so complex, for example, problems related to an airplane flying at transonic speeds or the effect of a blast wave on a building, that the traditional approach failed to yield useful information. These problems forced mathematicians, scientists, and engineers to turn their attention to numerical solutions obtained by computers. Of course, until the introduction of the digital computer in the early 1950s, the word computers referred to people (usually women) who performed

M.D. Salas (B) NASA Langley Research Center, Hampton, VA 23681, USA e-mail: [email protected]


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computations, not to machines. The following recollection by Godunov [10] is particularly apropos: [The calculations] were performed by a large group of exclusively female operators on Mercedes-brand electronic adding machines using the method of characteristics. . . The technique of these calculations was very well developed. The operators played on the keys of the calculators like pianists while actively discussing household matters and other problems that usually excite female interest. . . They often acidly remarked that our pseudoscientific explanations for the various numerical troubles which frequently hindered the calculations usually turned to be useless. One should keep in mind that they were paid for the volume of calculations, as determined by the number of filled-up lines in the tables. Any calculations with errors were not taken into account.

For a similar effort by the wives of Los Alamos National Lab scientists see the account of Harlow and Metropolis in [11]. Before the decade ended, the term “digital computer” gained usage to refer to the “supercomputers” of the day, such as the IBM 650, and distinguish them from their human counterpart. A remarkable example of the shift to numerical solutions is provided by Emmons’ work on transonic flows. From 1944 to 1948, under support from NACA,1 Emmons [7, 8] produced a series of transonic solutions that were at least 30 years ahead of their time. He solved the Euler equations in terms of a velocity potential and a stream function using the relaxation method of Southwell [25] in the subsonic region and a finite difference method of his own combined with a large doze of computational steering based on personal insight and good judgment in the supersonic region. Emmons describes his code for the subsonic region as follow (paraphrased): 1. Draw the airfoil and flow region to a scale such that the distance between net points is about 1.5 in. Do not use too many points at the start. 2. With the boundary conditions in mind, guess values of the stream function at the net points, and compute the residuals. To aid the accuracy of guessing, a freehand sketch of the streamlines and potential lines is somewhat useful. Use whole values of the stream function ranging from say 0 to 1,000. 3. The residuals are relaxed, each time recording at each point the change in stream function and the resultant residual. In this way the points at which the residual is largest can be spotted at a glance and relaxed next. 4. After all the residuals have values between ±2 add changes to the stream function to get final value at each point. 5. Re-compute the residual to locate any computation error. 6. If the solution is not accurate enough, additional points are added where needed. The code for fitting shocks is described as follows (paraphrased): 1. Solve problem as previously described in the absence of shocks. 2. A shock is arbitrarily placed in some location in the supersonic region.

1

National Advisory Committee for Aeronautics, created in 1915.

A Brief History of Shock-Fitting

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3. With this shock fixed the flow in the region following the shock is determined by the shock boundary conditions on the stream function and entropy distribution. 4. On completing this solution it will be found that the streamline directions following the shock do not agree with the assumed shock inclination. Change the shock inclination to get agreement and repeat step 3. 5. A few repetitions suffice to get an accurate solution. In the conclusion section to [7] Emmons adds (italics mine): “All of the methods described have one enormous advantage over analytical methods of solution of these problems. They permit the computer to use all of the facts he knows about the phenomena throughout the computations”. Emmons’ calculation of a compressible flow in a hyperbolic shaped channel was, to my knowledge, the first transonic calculation with a fitted shock wave, see Fig. 1. His other landmark calculation of the transonic flow over a NACA 0012 airfoil in free air and inside a tunnel, see Fig. 2, was not only the first transonic airfoil calculation with a fitted shock wave (see [20] for subsequent studies), but it also discovered a pressure singularity in the subsonic region at the foot of the shock. Emmons correctly explained the observed pressure behavior as follows: Since the stream is required to follow the airfoil surface, the curvature of the streamlines adjacent to the surface is specified by that surface. Since the surface is everywhere convex, there must be a pressure increase normal to the airfoil surface in order to cause the

Fig. 1 Adapted from fig. 10 of Emmons’ report [7]. Figure shows computed iso-Mach lines for flow through a hyperbolic channel. The calculated flow includes a fitted shock spanning the high of the channel near x = 0.5. This is the first known shock fitting result for a transonic problem

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Fig. 2 Adapted from fig. 5 of Emmons’ report [8]. This is first transonic airfoil calculation with a fitted shock. NACA airfoil inside a wind-tunnel, wind tunnel height is 1.8 chords, M∞ = 0.75 and zero angle of attack

velocity vector to turn as the fluid flows along the surface. Following the shock, however, the shock conditions have produced a pressure variation normal to the surface dependent on the pressure variation just prior to the shock, and in general for Mach numbers near 1 the pressure variation normal to the surface will be reversed by the shock. Hence the fluid must readjust itself very rapidly if it is to follow the airfoil surface. This curvature condition is responsible for the rapid pressure rise [8].

Many years later, Gadd [9] and then Oswatisch and Zierep [17] worked out the mathematical nature of the singularity showing that, immediately downstream of the shock, a logarithmically infinite acceleration develops along the surface. To better understand the magnitude of Emmons achievement, consider the effort to calculate an implosion at Los Alamos National Lab at around the same time. The sketchy accounts we have of these calculations are provided by an interview of Nicolas Metropolis by W. Aspray [2] and the article by Harlow and Metropolis [11]. In [11], the problem of computing a one-dimensional time-dependent implosion is described as follows: The problem of highest priority for the business machines was simulation of implosions, which involved integrating a coupled set of nonlinear differential equations through time from a prescribed initial configuration. The numerical procedure used a punch card for each point in space and time; a deck of cards represented the state of the implosion at a specific instant of time. Processing a deck of cards through one cycle in the calculation effectively integrated the differential equations ahead one step in the time dimension. This one cycle required processing the cards through about a dozen separate machines with each card spending 1 to 5 s at each machine.

The computer performed the repetitive calculations at each point (probably using a method of characteristics), however the calculation at the shock was done by hand at each time step and the deck of cards was modified accordingly before the next time step was started. Here is the account by Metropolis [2]:


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[The] shock-fitting had to be done external to the process and then punched in to the machine as the new initial condition for that operation, that particular cycle. . . . We had to punch out a set of cards that corresponded to the initial condition.

The process of shock-fitting was so disruptive that it compelled von Neumann and Richtmyer, who were working with the Los Alamos group, to introduced the idea of capturing shocks using an artificial dissipative term “to give shocks a thickness comparable to (but preferably somewhat larger than) the spacing of the points of the network” [27].2 The motivation for their idea was to simplify the numerical solution of the equations of fluid motion which is “usually severely complicated by the presence of shocks. . ., [because] the motion of the [shock] surfaces is not known in advance but is governed by the differential equations and boundary conditions themselves”. The great appeal of this approach is that it treats all points the same way (a sort of computational democracy), hence a local analysis of the scheme is globally valid (except at boundaries): in Moretti’s words “one code that can describe any flow” [14]. The paper by von Neumann and Richtmyer consisted of an analysis of the time dependent, one-dimensional Euler equations written in terms of the specific volume, the fluid velocity and the internal energy per unit mass with an additional artificial dissipation term of the form ∼ (x)2 u x |u x | added to the momentum equation. By doing what we today call a von Neumann stability analysis, they showed that both the modified differential equations and a proposed second order difference scheme were stable. They asserted that “the method [had] been applied, so far, only to one-dimensional flows, but [appeared] to be equally suited to study more complicated flows”. However, no results were presented. At the time of their report the need to express the equations in conservation form in order to conserve mass across the shock and capture the right shock jumps and speed had not been established. Thus, the equations they used were not written in conservation form and we suspect that the results they had did not showed the right shock speeds. The paper was published in 1950, but it is probably a good assumption that the method had been in use by von Neumann and Richtmyer for a few years during their secret testing of the ENIAC machine at the Ballistic Research Lab. When the NACA employees of the Langley Aeronautical Laboratory reported to work on Wednesday, October 1st, 1958, they drove past a new sign on the gate renaming the lab the NASA Langley Research Center. The creation of NASA was a direct consequence of a shreek-shreek-shreeeek heard around the world by radio operators from a Russian satellite, the size of a beach ball, launched the same week a year earlier. The transformation from NACA to NASA took place without fanfare, but like a sleepy volcano about to erupt U.S. aeronautical research was at the brink of a major resurgence. NASA and a number of other government agencies made investments to advance electronic computing and numerical analysis particularly for problems of interest to the space-race. Thus, NASA’s first technical report is a study by Van Dyke and Gordon of the supersonic blunt body problem [26]. The problem is very difficult to treat, other than numerically, because the shock is 2

See Godunov’s Reminiscences [1] for parallel developments in Russia.

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Fig. 3 Sketch of blunt body shock layer structure

an unknown boundary and the flow in the layer between the shock and the body changes type, being subsonic near the nose of the body and supersonic away from the nose, see Fig. 3. Three methods were popular in the late 1950s to early 1960s for solving this problem. All three fitted the shock wave as a boundary of the flow. One was the inverse method of Van Dyke [26], another was the direct integral relations method of Dorodnitsyn [6] implemented by Belotserkovskii [3], and finally the direct time-asymptotic method of Rusanov [21, 22] and Moretti [15]. Both the inverse and integral relations methods attempt to simplify the problem by casting it as a steady state problem. In Van Dyke’s formulation of the inverse method the problem is reduced to the numerical integration of two differential equations for the density and stream function. These are integrated proceeding downstream from an assumed shock wave shape towards the body, an approach that completely defies the elliptic character of the equations in the subsonic region. The equations are integrated until the stream function becomes negative. The body corresponding to the assumed shock shape is found by locating the zeroes of the stream function. The inverse method is ill-posed, since the shock shape is not very sensitive to changes in the body shape, and the whole approach is frail due to instabilities. Nevertheless, Van Dyke was able to overcome all these obstacles and obtained solutions for many body shapes of interest on one of the first general purpose electronic computers, the IBM 650.3 3 The 650 had only 2000 words of memory. Initially it was programmed in machine language, then in SOAP (Symbolic Optimal Assembly Program). By 1957 a FORTRAN compiler was available which compiled FORTRAN into SOAP.


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The time asymptotic method is so common today for solving steady state problems that it is not easy to grasp the mind-leap taken when the method was first introduced. But a leap it was in the early 1960s. With this method, the problem is framed as an initial-boundary value problem which is integrated in time until a steady state is reached. Using this approach the steady state problem which is a mixed elliptic-hyperbolic problem becomes strictly hyperbolic. Once the problem is formulated this way, the definition of initial conditions is perhaps the only challenge. The initial conditions require guessing at the shape of the bow shock and defining a shock layer flow field that is reasonably consistent with the governing equations and not too far from the steady state solution. If the initial conditions are too far from the steady state, then the path to the steady state might go through an intermediate flow structure different from that depicted on Fig. 3. Fortunately, the bow shock shape is not very sensitive to the body shape and there is a large experimental, theoretical and numerical knowledge-base on what the shock shape and flow shock layer should look like. The boundary conditions are relatively simple. The free stream is uniform and constant and, since the flow is supersonic, no signals propagate upstream. Therefore, only the values of the free stream immediately upstream of the bow shock are needed. These, together with the shock shape and Rankine-Hugoniot jumps, define the inflow boundary at the shock. Downstream of the sonic lines, the region of integration is terminated by outflow lines, see lines ab and cd on Fig. 3. The only criteria for selecting these lines are: (a) that they are sufficiently downstream for the flow to be fully supersonic, and (b) that they are sufficiently downstream from the limiting characteristic touching the sonic line, lines labeled on Fig. 3. Since these outflow boundaries lie in the supersonic region, extrapolation of flow conditions from inside the layer is valid. The boundary condition on the blunt body surface is of course the vanishing of the velocity component normal to the surface. In Moretti’s approach, the problem is formulated in polar or spherical coordinates depending on the problem being two-dimensional or three-dimensional. Considering only the two-dimensional case, the physical plane delimited by the lines ab, bd, dc, and ca is transformed to a rectangular computational plane by the coordinate transformation ζ =

r − rb(θ) , rs (t, θ ) − rb(θ) Y = π − θ, T = t,

where rb is the radial coordinate of the blunt body, rs is the radial coordinate of the bow shock, and r, θ are the polar coordinates. Grid points on the shock layer are computed using a modified Lax-Wendroff scheme, while body and shock points are computed by a method of characteristics. Understanding how the method of characteristics for the shock computation works is fundamental to shock-fitting. Thus we will explain it in some detail. To simplify the description, we do it in one space dimension. Two and three dimensions

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in space add geometrical complexity, but have no additional conceptual complexity. In one space dimension, the governing equations, expressed in characteristic form, are: λ± = u ± a, a P ± γ u = 0 on λ± , dx = u. d S = 0 on dt

(1)

The first of (1) is the characteristic slope. The second is the compatibility condition valid along the λ± characteristic expressed in terms of the natural logarithm of pressure, P = ln p, suitably normalized, and the velocity u, also suitably normalized. The primes denote differentiation along the characteristic. The isentropic exponent is denoted by γ . The third equation expresses the fact that the entropy, S, suitably normalized, is constant along the particle path. See reference [23] for details. Now consider Fig. 4. Assume that the flow is supersonic to the left of the shock and that it is subsonic on the right. At the starting time, t, the solution is known and we want to evaluate it at t + t. The new shock position, point b, is obtained by integrating the shock speed, w, in time: xb = xa + w(t)t. Second order accuracy can be achieved by using a predictor corrector scheme. At the shock point b all the flow conditions are known to the left of the shock, the supersonic side, either because this side corresponds to the free stream or because the values here are computed with a one-sided scheme using only values to the left of the shock.

Fig. 4 Details of shock computation by method of characteristics. Mesh points are the points labeled a, b and c. Point d is the origin of the λ− characteristic


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At point b, we have three Rankine-Hugoniot conditions: p2 = p1

2γ γ −1 2 − M , γ + 1 1r el γ + 1

2 (γ − 1)M1r u2 − w el + 2 , = 2 u1 − w (γ + 1)M1r el 2 M2r el =

2 (γ − 1)M1r el + 2 2 2γ M1r el − (γ − 1)

,

(2)

where the subscripts 1 and 2 refer to the supersonic and subsonic sides of the shock, respectively, and Mirel = (u i − w)/ai , i = 1, 2, is the Mach number relative to the shock. The Rankine-Hugoniot conditions are sufficient to find all the flow variables on the subsonic side of point b. However, to close the problem, we must also find the new shock speed w(t +t). The additional equation needed is obtained from the compatibility equation valid on λ− . This equation can be integrated to obtain: 1 (ab + ad )(Pb − Pd ) − γ (u b − u d ) = 0, 2

(3)

where values at d are interpolated from the values at a and c. The location of d, the foot of the characteristic, is given by: 1 x d = xb − (u b + u d − (ab + ad ))t. 2

(4)

Since the equations are nonlinear, an iterative process is used to solve them. This, or some variant of this, was used for fitting-shocks until the early 1970s.

2 Shock-Fitting Matures The shock-fitting method described for the blunt body problem is known as boundary shock-fitting. It reached a high level of maturity in the work of Marconi and Salas [16]. In this work, the boundary shock-fitting method was used to solve problems typical of high speed flow over aircraft- and spacecraft-like configurations. Figure 5, adapted from [16], illustrates the level of complexity that was achieved. The configuration represents a high speed research aircraft that was of interest to NASA at that time. The flight conditions correspond to M∞ = 6, γ = 1.2 and zero angle of attack. In the simulation, the aircraft bow shock, canopy shock, wing shock and vertical tail shock are treated as boundaries. The solution of the Euler equation required approximately one hour of CPU time on an IBM 370/168 computer using a cross sectional grid consisting of 25 by 30 mesh points. This computer was among

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Fig. 5 Shock pattern calculated for a high speed research aircraft. Free stream conditions corresponding to M∞ = 6, γ = 1.2 and zero angle of attack, original figure from [16]

the fastest computers available in the early 1970, running at about 3.5 MIPS (million instructions per second). It became evident with this work that the technique of creating shock-bounded regions was a limiting factor for multi-dimensional complex flows. This led Moretti to develop a technique where “the shocks float among mesh points” [13] which he labeled floating shock-fitting. For one-dimensional problems, floating shock-fitting has the additional advantage over boundary shock-fitting in that it does not require adding and subtracting mesh points as regions enlarge or contract. This is not just the cost in additional logic, but also the elimination of interpolation errors associated with the addition and subtraction of mesh points. The disadvantage of floating shock-fitting, however, is the need for additional difference expressions for mesh points in the vicinity of shocks (both in space and time) and the possibility of inconsistencies between the truncation errors from these expressions and those of the numerical scheme used for the rest of the field. An example of what could be accomplished with floating shock-fitting is shown in Fig. 6 depicting a 1976 computation of a simulated scramjet [24]. The method of characteristics used for the shock evaluation remained an idiosyncrasy until 1970 when Czeslaw Kentzer [12], a professor at Purdue University, found


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Fig. 6 Flow field for a simulated scramjet, inflow Mach = 3.0, angle of incidence 8◦ . Figure shows shock waves (heavy solid lines), contact sheets (dashed lines), isobars (solid lines), original figure from [24]

a way around it. Kentzer realized that what was needed was a partial differential equation for the shock acceleration, dw/dt. He obtained this equation as follows. Since the Rankine-Hugoniot conditions are valid on the shock surface, introduce a local coordinate transformation, say X, T, which makes the shock surface line up with one of the new coordinates, for example let X = 0 corresponds to the shock surface. Now differentiating (2) with respect to T , we find 4γ p1 M1r el M1r elT , γ + 1 p2 a2 M2r el 4a1 u 2T = wT + (u 1T − wT ) − M1r elT , 2 a1 M1r el (γ + 1)M1r el ai T u i T − wT Mir elT = − Mir el , i = 1, 2, ai ai γ −1 Si T ai T = ai Pi T + , i = 1, 2. 2γ γ −1 P2T = P1T +

(5)

The compatibility condition along the contravariant characteristic direction, defined − by − 2 = X t + λ2 X x , is − a2 P2T + − 2 P2X − γ u 2T + 2 u 2X = 0. Combining (6) and (5), then solving for wT , we find the desired equation: wT = (B + C)/A

(6)

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where A = γ (E + D − 1), − B = a2 P1T + − 2 P2X + γ Eu 1T + 2 u 2X , C = γ D(u 1T − M1r el a1T ), 1 4 a2 p 1 , M1r el − 2 D= γ + 1 a1 p2 M1r el E=

(7)

a2 M2r el . a1 M1r el

The terms B and C in (7) depend on the derivatives P1T , u 1T , and a1T . If the low pressure side corresponds to a constant state, such as a free stream, these derivatives would be zero; otherwise, these derivatives are evaluated using only information on the left side of the shock. The other derivatives in the term B are P2X and u 2X . These derivatives provide information about the local environment on the high pressure side of the shock. They are evaluated with one-sided differences using only values on the right side of the shock. The Kentzer approach can be simplified by recognizing that the information carried to the shock by the λ− characteristic is encapsulated in the Riemann variable Q=

2a − u. γ −1

It follows from the Rankine-Hugoniot jumps that on the right side of the shock this Riemann variable is defined by Q2 = u1 +

2a1 g(M1r el ), γ +1

(8)

where g(M1r el ) =

1 2 2 2 (2γ M1r el − (γ − 1))((γ − 1)M1r el + 2)/M1r el γ −1 2 + (M1r el − 1)/M1r el .

Differentiating (8) with respect to T , we find the following equation for the shock acceleration γ +1 γ +1 wT = 1 + u 1T − Q 2T , 2g 2g where g = ∂g/∂ M1rel . Ideally, Q 2T should be evaluated with an upwind scheme written in terms of Riemann variables, such as the λ-scheme [14]. However, numerical experiments show that it also works rather well even with central difference


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schemes. For the extension of this method to multi-dimensions and for a comparison of numerical results of different shock-fitting methods see [23]. Every shock-fitting method is build around knowledge of the shock jump conditions. Since the classical equations of fluid mechanics consist of a set of conservation equations that can be written in divergence form, the shock jumps are easily obtained and shock-fitting methods can be formulated. However, this is not true for many other problems in mathematical physics which are expressed by equations that cannot be written in divergence form. How do we treat shock waves for these problems? The answer does not lie in turning to shock capturing methods as these methods have their own problems [1]. The answer lies in formulation of a mathematical theory for dealing with the multiplication of distribution functions. Such a theory was developed by J. F. Colombeau and co-workers in the 1980s and it is now well developed [5, 23]. Space does not allow me to go into the details here, but suffices to say that methods exist to obtain shock jumps for equations that cannot be expressed in divergence form and shock-fitting methods are easily applied to these equations [23].

3 New Developments in Shock-Fitting In the late 1940s, von Neumann and researchers at Los Alamos National Lab struggled to formulate a shock-fitting method for a relatively simple one-dimensional time dependent problem with the tools available then. Progress in numerical analysis, computer languages and hardware made possible the numerical application of shock-fitting methods to much more difficult problems, but it is undisputable that shock-fitting methods are not yet sufficiently developed to handle general threedimensional problems with complex shock interactions. However, with the maturity of both upwind methods and unstructured grid techniques, the future of shock-fitting looks very promising. In computational fluid dynamics, research in unstructured grids techniques was started in the mid 1980s to deal with the geometrical complexities of realistic aircraft configurations. Unstructured grid methods require a significant amount of infrastructure to deal with almost random grid point connectivity, fast re-gridding and bookkeeping of complex data structures, but these are just the ingredients needed for the development of shock-fitting codes for complex three-dimensional problems. Taking advantage of the infrastructure of unstructured grids, Aldo Bonfiglioli and Renato Paciorri [18] have in the last few years developed a floating shock-fitting capability that appears suitable for problems with very complex shock patterns. Their method in two space dimensions consists of a tessellation of the computational plane based on a constrained Delaunay triangulation. This triangular mesh is treated as a background mesh over which the shock fronts move. The numerical scheme used is based on a cell-vertex, upwind, conservative algorithm that can be used in both a shock-capturing and/or a shock-fitting mode. This capability has two important consequences: (1) it allows for direct comparison of shock-capturing and shock-

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fitting results; and (2) if, for some reason, a shock is not fitted, a shock-capturing solution is obtained in its place. In this method, the location of shock-nodes is stored on a table. Since, in general, the background mesh is not suitable for fitting the shock front, at the beginning of every computational step a constrained Delaunay triangulation re-meshing of the mesh is performed to locally align the mesh with the shock. With the shock front now forming an internal boundary of the mesh, a shock-fitting method is used to solve for the new shock position and to evaluate the flow variables on the high pressure side of the shock. Once the solution is updated on the locally shock-aligned mesh, this information is used to update the mesh points on the background mesh near the shock. Thus, at the end of a computational step a new shock position is known, a flow field consistent with the Rankine-Hugoniot jumps is known at the shock nodes, and the flow field on the background mesh nodes, consistent with the fitted shock front, is also known. The details can be found in [18] and [23]. Figures 7 and 8 show results obtained by Bonfiglioli and Paciorri [19] for a type IV shock interaction between an incident planar shock and the bow shock of a circular cylinder. The free stream Mach number is 10 and the incident shock is inclined at 10◦ to the free stream. Figure 7 corresponds to a partial shock-fitting solution, while in Fig. 8 all shocks are fitted. The significant improvement on the quality of the solution with the full shock-fitted calculation is obvious. Conceptually, the extension of this method to three-dimensional flows is equivalent to the two-dimensional case. However, the implementation is far from triv-

Fig. 7 Computed Type IV shock interaction. The bow shock and the shock connecting the two triple points are fitted; all other discontinuities are captured. Right panel shows details around shock interaction. Results presented courtesy of R. Paciorri [19]


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Fig. 8 Computed Type IV shock interaction using fully shock-fitting mode. Right panel shows details around shock interaction. Results presented courtesy of R. Paciorri [19]

ial. The constrained Delaunay triangulation, used in the two-dimensional tessellation, is now replaced by a constrained Delaunay tetrahedralization. However, while it is always possible to build a constrained Delaunay triangulation in twodimensions, the same is not true for a constrained Delaunay tetrahedralization of three-dimensional space. This problem can be overcome with the addition of Steiner points4 or by carefully defining the extent of the region to be re-meshed. The first effort in this direction can be found in [4].

4 Conclusion In the early days of computational fluid dynamics, the only way of treating shock waves was through shock-fitting which had its roots in the method of characteristics. Early researchers, however, found shock-fitting cumbersome and looked for methods to treat shocks in a way homogeneous with the other field points. This eventually resulted in the now popular shock capturing method. A few researchers continued the development of shock-fitting methods demonstrating its superior accuracy and efficiency over shock capturing methods. These shock-fitting methods, however, never matured to the point of being able to handle general highly complex shockproblems. The Achilles’ heel of shock-fitting has been the difficult bookkeeping of the complex shock structures that develop in two and three space dimensions. 4

A node not in the original grid added to improve the quality of the tezzellation.

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However, with the maturity now achieve with upwind unstructured grid techniques shock-fitting methods appear to be experiencing a renaissance.

References 1. Abgrall, R., Karni, S.: A comment on the computation of non-conservative products. J. Comput. Phys. 229(8), 2759–2763 (2010) 2. Aspray, W.: An interview with Nicolas C. Metropolis. Charles Babbage Institute, The Center for the History of Information Processing, University of Minnesota, Minneapolis, http:// special.lib.umn.edu/cbi/oh/pdf.phtml?id$=$198 (1987) 3. Belotserkovskii, O.M.: On the numerical calculation of detached bow shock waves in hypersonic flow. Prikl. Mat. Mekh. 22, 206–219 (1958) (Trans: J. Appl. Math. Mech. 22, 279–296) 4. Bonfiglioli, A., et al.: An unstructured, three-dimensional, shock-fitting solver for hypersonic flows. AIAA Paper 2010–4450, 40th Fluid Dynamics Conference and Exhibit, 28 June – 1 July 2010. Chicago, Illinois (2010) 5. Colombeau, J.F.: Elementary Introduction to New Generalized Functions. North-Holland Mathematics Studies 113, North-Holland, Amsterdan (1985) 6. Dorodnitsyn, A.A.: On a method of numerical solution of some nonlinear problems of aerohydrodynamics. In: Proceedings of the 9th International Congress of Applied Mechanics, vol. 1, p. 485. University of Brussels, Brussels (1957) 7. Emmons, H.W.: The Numerical Solution of Compressible Fluid Flow Problems, NACA Tech. Note no. 932 (1944) 8. Emmons, H.W.: Flow of a Compressible Fluid Past a Symmetrical Airfoil in a Wind Tunnel and in Free Air, NACA Tech. Note no. 1746 (1948) 9. Gadd, G.E.: The possibility of normal shock waves on a body with convex surfaces in inviscid transonic flow. Zeit. Ang. Math. Phys. 11, 51–55 (1960) 10. Godunov, S.K.: Reminiscences about difference schemes. J. C. P. 153, 6–25 (1999) 11. Harlow, F.H., Metropolis, N.: Computing & Computers, Weapons Simulation Leads to the Computer Era. Los Alamos Science, Winter/Spring. http://www.lanl.gov/history/hbombon/ pdf/00285876.pdf (1983) 12. Kentzer, C.: Discretization of boundary conditions on moving discontinuities. In: Proceedings of the II International Conference on Numerical Methods in Fluid Dynamics, pp. 108–113 (1970) 13. Moretti, G.: Experiments in Multi-Dimensional Floating Shock-Fitting. PIBAL report no. 73–18. Polytechnic Institute of Brooklyn, Brooklyn, NY (1973) 14. Moretti, G.: Computation of flows with shocks. Ann. Rev. Fluid Mech. 19, 313–337 (1987) 15. Moretti, G., Abbett, M.: A time-dependent computational method for blunt body flows. AIAA J. 4, 2136–2141 (1966) 16. Marconi, F., Salas, M.D.: Computation of three-dimensional flows about aircraft confingurations. Comput. Fluids. 1, 185–195 (1973) 17. Oswatitsch, K., Zierep, J.: Das Problem des Senkrechten Stosses an einer gekrümmten. Wand. Zeit. Ang. Math. Mech. 40(Supp), 143–144 (1960) 18. Paciorri, R., Bonfiglioli, A.: Numerical simulation of shock interaction with unstructured shock-fitting technique.In: Sixth European Symposium on Aerothermodynamics of Space Vechicles, Paris, France (2008). 19. Paciorri, R., Bonfiglioli, A.: A shock-fitting technique for 2D unstructured grids. Comp. Fluids. 38, 715–726 (2009) 20. Rizzi, A., Viviand, H. (eds.): Numerical Methods for the Computation of Inviscid Transonic Flows with Shock Waves, Notes on Numerical Fluid Mechanics, vol. 3, Friedr. Vieweg & Sohn, Braunschweig (1981)


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21. Rusanov, V.V.: A three-dimensional supersonic gas flow past smooth blunt bodies. In: Proceedings of the 11th International Congress of Applied Mechanics, pp. 774–778 (1964) 22. Rusanov, V.V.: A blunt body in a supersonic stream. Ann. Rev. Fluid Mech. 8, 377–404 (1976) 23. Salas, M.D.: A Shock-Fitting Primer. Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Hampton, VA (2009) 24. Salas, M.D.: Shock fitting method for complicated two-dimensional supersonic flows. AIAA J. 14, 583–588 (1976) 25. Southwell, R.V.: Relaxation Methods in Engineering Science. Oxford University Press, Oxford (1940) 26. Van Dyke, M.D., Gordon, H.D.: Supersonic Flow Past a Family of Blunt Axisymmetric Bodies, NASA TR R-1 (1959) 27. von Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. App. Phys. 21, 232–237 (1950)

Understanding Aerodynamics Using Computers Mohamed M. Hafez

Abstract It is argued that Aerodynamics Education can benefit from using elementary CFD codes, at least for benchmark problems. To understand how forces and moments are generated on bodies moving in air, the dependence of surface pressure and skin friction on models, media and motion must be examined theoretically, experimentally or numerically. The non-dimensional geometry parameters of wings are the thickness, camber and aspect ratios as well as the angle of attack, while the parameters of the motion are Reynolds, Mach and Strouhal numbers (assuming the Prandtl number and the ratio of specific heats for the air are constants under operating conditions). Obviously, the theory is limited to simple cases, where solutions are available, while experiments are in general expensive. Hence, CFD became a useful tool to study aerodynamics. Sophisticated and efficient codes are used nowadays in Industry for both analysis and design processes. Over the last four decades, numerical methods have been developed to understand aerodynamic phenomena as well as to accurately predict aerodynamic loadings for practical applications. These codes can be used also in Academia for teaching purposes. However, students will not be able to easily modify these codes and such black box tools are not ideal for engineering education. It appears, the basic aerodynamic theories, including three dimensional, compressibility and viscous effects, as well as standard numerical methods with simple grids, are to be used in order to produce the main results in a straight-forward manner. Students should be able to write their own codes from scratch, using finite difference or finite volume schemes (with artificial viscosity), and solvers based on Gaussian elimination of block triadiagonal systems of algebraic equations. The dependence of lift, drag, moment coefficients on geometry and motion parameters can be produced numerically for benchmark problems, where theoretical solutions are available, and parametric studies are feasible on small personal computers. M.M. Hafez (B) Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616-5294, USA e-mail: [email protected]


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The students must be, however, familiar with undergraduate linear algebra and programming language. On the other hand, the mathematics needed for obtaining the theoretical results are sometimes complicated, even for linear problems. For example, after the derivation of the potential equation and the corresponding boundary conditions under the assumptions of inviscid and irrotational flows, mathematical techniques like eigenfunction expansion, Green’s function or transform methods will be replaced by simple numerical methods based on finite differences. The classical Prandtl lifting line theory requires the solution of singular integro-differential equation! Even two dimensional incompressible flows require functions of complex variables, conformal mapping and residue theorem. Two dimensional supersonic flows require method of characteristics. Hankel functions and Laplace Transform are used in unsteady flow analysis. Transonic flows are governed by nonlinear equations and Hodograph transformation is limited to two dimensional smooth flows where boundary conditions become complicated. For viscous effects, Prandtl boundary layer equations can be easily solved numerically together with reliable viscous/inviscid interaction procedures. Alternative methods can also be used, and the results can be compared with those of triple deck theory. The ideas outlined above demonstrate how to use simple standard numerical techniques to reproduce the main results of aerodynamic theories which usually require sophisticated mathematical tools.

Part III

High-Order Methods

A Unifying Discontinuous CPR Formulation for the Navier–Stokes Equations on Mixed Grids Z.J. Wang, Haiyang Gao, and Takanori Haga

Abstract A unifying discontinuous formulation named the correction procedure via reconstruction (CPR) for conservation laws is extended to solve the Navier–Stokes equations for mixed grids. The CPR framework can unify several popular high order methods including the discontinuous Galerkin and the spectral volume methods into a differential formulation without explicit volume or surface integrations. Several test cases are computed to demonstrate its performance.

1 Introduction The last two decades have witnessed considerable research interest for adaptive high-order methods in the computational fluid dynamics (CFD) community, e.g., the discontinuous Galerkin (DG) [2, 7], the staggered-grid (SG) [5], spectral volume (SV) [9], and spectral difference (SD) [6] methods. A review of high-order methods on unstructured meshes was given in Ref. [10]. Recently, a novel formulation named CPR (correction procedure via reconstruction) was developed by Huynh [4] for 1D conservation laws, and extended to simplex and hybrid meshes by Wang and Gao [11]. The CPR formulation is based on a nodal differential form, with an element-wise continuous polynomial solution space. The framework is easy to understand, efficient to implement and recovers several known methods such as the DG, SG or the SV/SD methods. Furthermore, by choosing the solution points to coincide with the flux points, the reconstruction of solution polynomials to calculate the residual can be avoided. In a present paper, we present the CPR framework for the Navier–Stokes equations on mixed meshes. The basic formulation of the CPR method for the compressible Navier–Stokes equations is presented in the next section. Section 3 presents the computational results for several test problems. Conclusions for the present study and possible future work are summarized in Sect. 4. Z.J. Wang (B) Department of Aerospace Engineering, CFD Center, Iowa State University, Ames, IA 50011, USA, e-mail: [email protected]


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2 The CPR Formulation for the Navier–Stokes Equations The compressible Navier–Stokes equations can be written as a system of partial differential equations in conservation form

∂Q Q = 0, · Fc (Q) − Fv Q, ∇ +∇ ∂t

(1)

where Q, Fc and Fv denote the conservative state vector, the inviscid and the viscous flux vectors, respectively. In order to discretize the Navier–Stokes equations, we follow a mixed formulation that is commonly used in the DG method [1, 3]. By Q, Eq. (1) is rewritten as a first order system introducing a new variable R = ∇

∂Q + ∇ · Fc (Q) − Fv Q, R = 0, ∂t Q. R = ∇

(2) (3)

Let P k (Vi ) denote the space of degree k or less polynomials on element Vi . Assume that the numerical approximations of Q and R on Vi are Q i , Ri , and they belong to P k (Vi ). A nodal set { ri, j } Kj=1 called solution points (SPs) as shown in Fig. 1a are defined on element Vi with K being the dimension of P k (Vi ). The solutions at the SPs {Q i, j } Kj=1 can be used to form the solution polynomial using a Lagrange interpolation

Qi =

K

Q i, j L j ( r ),

(4)

j=1

r ) is the Lagrange polynomial. The CPR formulation applied to (2) and where L j ( (3) can be written as

∂ Q i, j Fc (Q i ) − Fv (Q i, , Ri ) + 1 α j, f,l + ∇· j |Vi | ∂t f ∈∂ Vi l

Fcn f,l − Fvn f,l S f = 0, Qi ) j + 1 Ri, j =(∇ α j, f,l [Q] f,l n f S f , |Vi | f ∈∂ Vi

(5) (6)

l

where |Vi | is the volume of Vi , S f , is the area of face f, α j, f,l are constant lifting coefficients, and

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(a)

61

(b)

Fig. 1 Efficient arrangement of solution (squares) and flux points (circles) for k = 2

n (Q i , Q i+ , n) − Fcn (Q i ), Fcn ≡ Fc,com

n (Q i , Q i+ , ∇ Q i , ∇ Q i+ , n) − Fvn (Q i , Ri ), Fvn ≡ Fv,com [Q] ≡ Q com (Q i , Q i+ ) − Q i ,

n n with Fc,com the common normal inviscid flux, Fv,com the common normal viscous flux, and Q com the common state variable at a face, Q i+ the solution outside element Vi . The method for the quadrilateral element is one-dimensional in each direction as shown in Fig. 1b. We need to compute the internal flux divergence and the common flux at the interface in (5). Instead of approximating the inviscid flux by the Lagrange interpolation on the SPs, the flux divergence is calculated “exactly” at the solution points with the chain rule (CR) approach

· Fc (Q i, j ) = ∇

∂ Fc (Q i, j ) Q i, j , ·∇ ∂Q

(7)

· Fc (Q i ) is generally not where ∂∂ FQc is the inviscid flux Jacobian matrix. Note that ∇ a degree k polynomial, but it can be approximated by the Lagrange polynomial of degree k using the flux divergence at the solution points, i.e., · Fc (Q i ) ≈ ∇

· Fc (Q i, j ), L j ( r )∇

(8)

j

The common inviscid flux can be obtained with any Riemann solver. In this paper, the Roe flux [8] is used for all the cases. In the present study, we employ the BR2 scheme [1] to discretize the viscous flux. In (6), the common solution Q com, f,l is simply the average of the solutions at both sides of f . The viscous fluxes at the solution points are evaluated with F v (Q i, j , Ri, j ). Then the viscous flux divergence is obtained through the Lagrange interpolation rather than the CR approach [11]. The common viscous flux also needs

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to be determined. Besides the common solution, we also need to define a common gradient on face f . The common gradient is evaluated as

1 − − + + Q com = + r + ∇ Q + r ∇ ∇ Q f,l f,l f,l f,l f,l , 2

(9)

Q + are the gradients of the solution from the left and right Q − and ∇ where ∇ f,l f,l + cells, while r − and r f,l f,l are the local lifting correction to the gradients only due to the common solution on face f 1 r± αl, f,m [Q]±f,m ∓ n f Sf, f,l = ± V

(10)

m

where m is the index for the flux points on f . Note that there is no summation over all faces of the element in Eq. (10) in order to assure local property of the BR2 scheme.

3 Numerical Results 3.1 Boundary Layer over a Flat Plate The Reynolds number based on the plate length L is Re L = 10, 000 and the freestream Mach number is M = 0.2. The computational grid has 4 cells in the boundary layer at x/L = 1.0 and 13 cells along the plate. The computed v velocity profile at x/L = 0.5 is compared with the Blasius’s solution in Fig. 2a, while the computed skin friction coefficients on the wall are plotted in Fig. 2b. It is clear that the agreement with the Blasius’s solution becomes better with p-order refinement. (a)

(b) 0.1

6

Blasius 2nd-order 3rd-order 4th-order

Blasius 2nd-order 3rd-order 4th-order

5

C

f

4 3 2 0.01

1 0 –0.2

0

0.2

0.4

0.6

0.8 1/2

v(2*Re*x) /U

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

x

Fig. 2 Comparisons of v-velocity profile at x/L = 0.5, and the skin friction distribution along the wall

A Unifying Discontinuous CPR Formulation for the Navier–Stokes Equations 6 5

63

Blasius 3rd-order (~8 cells in the BL) 3rd+6th-order (~2 cells in the BL)

4 3 2 1 0 –0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1/2

v(2*Re*x)

/U∞

Fig. 3 Comparison of v-velocity profiles using different degrees of polynomial and grids

The CPR formulation can easily allow different polynomial orders in different directions. Such an example is shown in Fig. 3. In the wall normal direction, a degree 5 polynomial was used on a coarse grid, and produced more accurate results than a degree 2 polynomial on a finer mesh.

3.2 Unsteady Subsonic Flow over a Sphere at Re = 300 We consider an unsteady flow case over a sphere with Reynolds number of 300 based on the diameter. The inflow Mach number is assumed to be 0.3. The computational mesh is shown Fig. 4. To resolve the shedding vortices, the mesh is generated to have fine cells in the wake region. The total number of mixed cells is 54,312.

Fig. 4 Computational grid around a sphere for the unsteady viscous flow over a sphere (left: entire grid, right: grid around the sphere)

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Fig. 5 Computed Q isosurface in the wake region at Re = 300

The computed Q isosurface colored by local Mach number using the 4th-order CPR scheme is shown in Fig. 5. The obtained plain symmetric wake vortex structure is comparable to other experimental and computational results.

4 Conclusion The CPR formulation has been successfully extended to solve the Navier–Stokes equations on hybrid unstructured meshes. The resulting scheme needs no explicit integrations and no data reconstructions. This numerical efficiency is more significant in 3D simulations in comparison to 2D simulations because numerical complexities involved in high-order quadratures and reconstructions rapidly increase in 3D. Numerical results indicated that the method performs satisfactorily for a wide variety of flow conditions. Future work includes the development of low-storage solution algorithms and hp-adaptation techniques. Acknowledgements This study has been supported by the Air Force Office of Scientific Research (AFOSR) under grant FA9550-09-1-0128.

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References 1. Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131(1), 267–279 (1997) 2. Cockburn, B., Shu, C.-W.: TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework. Math. Comput. 52, 411–435 (1989) 3. Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin methods for time-dependent convection diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998) 4. Huynh, H.T.: A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods. AIAA paper 2007–4079 5. Kopriva, D.A., Kolias, J.H.: A conservative staggered-grid Chebyshev multidomain method for compressible flows. J. Comput. Phys. 125, 244 (1996) 6. Liu, Y., Vinokur, M., Wang, Z.J.: Discontinuous spectral difference method for conservation laws on unstructured grids. J. Comput. Phys. 216, 780–801 (2006) 7. Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory Report, LA-UR-73-479 (1973) 8. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schems. J. Comput. Phys. 43, 357–372 (1981) 9. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys. 178, 210–251 (2002) 10. Wang, Z.J.: High-order methods for the Euler and Navier-Stokes equations on unstructured grids. J. Prog. Aerosp. Sci. 43, 1–47 (2007) 11. Wang, Z.J., Gao, H.: A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids. J. Comput. Phys. 228, 8161–8186 (2009)

Assessment of the Spectral Volume Method on Inviscid and Viscous Flows Oussama Chikhaoui, Jérémie Gressier, and Gilles Grondin

Abstract The compact high-order ‘Spectral Volume Method’ (SVM, Wang, J. Comput. Phys. 178(1):210–251) designed for conservation laws on unstructured grids is presented. Its spectral reconstruction is exposed briefly and its applications to the Euler equations are presented through several test cases to assess its accuracy and stability. Comparisons with classical methods such as MUSCL show the superiority of SVM. The SVM method arises as a high-order accurate scheme, geometrically flexible and computationally efficient.

1 Introduction Despite the constant improvements in computational and data processing resources, the continuously growing requirements of computational fluid dynamics still remain unsatisfied. In the last decade, the CFD community showed a growing interest in high-order approximations to address these issues (WENO, Discontinuous Galerkin, . . . ). An attractive choice is the Spectral Volume Method (SVM) proposed and developed by Wang et al. [3, 5, 6] which achieves high-order accuracy on unstructured grids through polynomial reconstruction. To assess the performance of the SVM method, different test cases were computed with Typhon, an unstructured open-source code. The numerical experiments were chosen to cover a large set of flow configurations from continuous quasiincompressible problems to shock wave propagations and mixing flows. The results presented here are up to expectations with a significant increase in accuracy and a reduction in CPU time compared to a usual second order method.

O. Chikhaoui (B) Département d’Aérodynamique, Énergétique et Propulsion, Institut Supérieur de l’Aéronautique et de l’Espace, 10 av. Edouard Belin - BP 54032 – 31055, Toulouse, France e-mail: [email protected]


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2 Spectral Volume Method for the 2D Euler Equations The SVM method achieves high-order accuracy on unstructured grids through polynomial reconstruction within initial grid cells (spectral volumes, SVs) with a subdivision of the each SV into polygonal control volumes (CVs, Fig. 1). The spectral splitting of the SV is designed to minimize internal reconstruction oscillations [1]. The flux computations are finally achieved with Gaussian quadrature directly inferred from CV states ponderation with a constant and unique set of coefficients for each SV in the whole domain. While high-order reconstructions are often based on projections and gradient evaluations on a quite large region of neighboring cells, the SVM compact stencil makes it computationally efficient and attractive. The polynomial reconstruction used remains exact (at a given order) on arbitrarily shaped triangles.

2.1 Spectral Volumes Partitions The subdivision of a spectral volume Si in different control volumes Ci j is an essential step for the SVM scheme. For example, some geometric partitions can C

B

A C

C

A1 O

R

L F A

D

G B A

D

Fig. 1 Geometric splitting definitions for the second, third and fourth order partitions

B

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lead to degenerated systems and thus should be excluded. The splitting procedure must conserve the existing symmetries in a triangle, use straight edges and consider convex CVs. Figure 1 shows different geometric partitions of a spectral volume for different desired orders. While the second order splitting definition is unique, the higher order partitions introduce some geometric parameters. The third order AF partitions is defined by two parameters: α = AD AB and β = A A1 . The fourth order AD AG partitions have four degrees of freedom: α = AB , β = A R , γ = OA RR and δ = AALR . The accuracy and stability of a given spectral volume scheme depends on the choice of these geometric parameters [1, 2]. Table 1 sums up the different implemented and tested partitions with their geometric parameters. The SVMW splittings were proposed by Wang et al. [6] and designed by minimizing the Lebesgue constant over the SV. Whereas the minimization of this constant provides a good assessment of the quality of the SVM splitting, different observations show that it is not a sufficient condition for the stability of the scheme. Thus, Abeele proposed other geometric splittings noted as SVMK and SVMK2 [1, 2].

2.2 Spectral Volume Method Assets The spectral volume method uses a compact stencil which is a great advantage compared to other high order reconstructions. The SVM geometric splitting is designed and optimised in a spectral way to minimize internal reconstruction oscillations known as Runge phenomena. The reconstructed field is continuous over the entire SV, therefore internal faces do not constitute Riemann problems, which reduces the flux computation cost and contains the problem of data limitation to the SV faces. The other interesting aspect of the SVM is the homothetic nature of the splitting reconstruction: no new metric terms need be kept in memory. The interpolation on Gauss points for fluxes computation is directly inferred from a weighting of CV states with constant and unique coefficients for the whole domain. Lastly, while the usual finite-volume and finite-difference methods depend strongly on the grid quality and density, the SVM reconstruction remains exact (at a given order) on arbitrarily shaped triangles.

Partition

Table 1 Geometric parameters for the different SVM partitions Order α β δ

γ

SVM2 SVM3W SVM3K SVM3K2 SVM4W SVM4K SVM4K2

2 3 3 3 4 4 4

– – – – 1/15 351/1000 0.1562524902

– 1/4 91/1000 0.1093621117 1/15 78/1000 0.0326228301

– 1/3 18/100 0.1730022492 2/15 104/1000 0.042508082

– – – – 2/15 52/1000 0.0504398911

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3 Numerical Experiments To assess the performance of the SVM method, different test cases were computed with Typhon, an open-source unstructured solver [4]. The numerical experiments were chosen to cover a large set of flow configurations. In the next sections, we focus on the results of two chosen cases: the evolution of a convected vortex and a simple Mach reflection on a wedge. The HLLC solver is used for the Riemann problem at faces. Time integration is performed with a third order TVD Runge-Kutta scheme and SVM results are compared to results provided by a usual MUSCL method.

3.1 Convected Vortex We consider a vortex evolution problem governed by: ∂∂tp = ρ vrθ and convected with a speed of Vconv = 20 in the x direction on a regular domain of [−5 : 5]×[−5 : 5]. Periodic boundary conditions are set in the x and y directions. Thus, the vortex crosses the whole domain twice from left to right between t = 0 and t = 1. No limiters were employed for the SVM simulations. The overview of the vortex damping is presented on Fig. 2 for the MUSCL simulation using the Van Albada limiter. While the theoretical maximum transverse velocity is V ymax = 30, Vymax ∼ 10 is obtained for this MUSCL simulation, thereby showing that this test case is very sensitive to numerical dissipation. For comparison purposes, the velocity profiles are considered on Fig. 3 at t = 1 along a horizontal line passing through the vortex center. With the SVM schemes, the velocity profile is better preserved when the order is increased. Thus, Vymax ∼ 15, Vymax ∼ 25 and Vymax ∼ 30 are obtained for the second, third and fourth order respectively. 2

Fig. 2 Overview of numerical damping of a convected vortex using MUSCL scheme

Assessment of the Spectral Volume Method on Inviscid and Viscous Flows 30

MUSCL SVM2 SVM3 SVM4 Exact

1 0.8

20

Ec/Ec_0

Velocity_Y

25

15 10

0.6 0.4

5 0

0.2 0

1.25 X

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2.5

MUSCL SVM2 SVM3 SVM4 Exact

0.2

0.4 0.6 Time

0.8

1

Fig. 3 Comparison of velocity profiles along a line through the vortex center at t = 1 (left) and kinetic energy time evolution (right)

Another interesting aspect of SVM simulations is the reduction in CPU time per time step compared to a classic MUSCL method using the same number of control volumes: 50% for 2nd order, 32–35% for 3rd order and 22–25% for 4th order. The CPU time savings are due to the absence of gradient evaluation for inviscid fluxes and the continuity of state variables through internal faces. This latter fact reduces the limitation problems and several cases can be computed without any limitation procedure.

3.2 Simple Mach Reflection This case deals with a classic problem of shock reflection with a Mach number Ms = 1.7 and a wedge angle of θ = 25◦ . The numerical results were obtained on a domain of [25 × 16.5] on the x, y plane with the apex of the wedge placed at x, y = 4.69, 0. The upstream shock conditions are ambient conditions, with ρa = 1.225, pa = 1.01325 · 105 and u = 0. Different SVM simulations were carried out with different orders. Observations show that the contours around the incident and reflected shocks and the slip surface are better captured when the order increases. To prove that the capture of these features and the better resolution of the flow are due to the increasing order, other simulations have been performed using the MUSCL scheme. The results obtained with the SVM4 scheme were compared to those provided by the MUSCL method on a finer structured cartesian grid. While the unstructured grid for SVM4 contains 422.080 CVs, the structured grid used for MUSCL simulation contains almost twice the number of CVs, i.e. 800.000 CVs. Yet, the shocks thicknesses obtained are large and no improvement of the slip surface resolution is observed when increasing the number of CVs with the MUSCL method. Figure 4 clearly shows that the SVM scheme better captures the slip surface: Kelvin-Helmholtz instabilities can be

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Fig. 4 Entropy contours at slip surface: MUSCL, 800.000 CV (left) and SVM4, 422.080 CV (right)

observed on the shear line. These observations suggest that the better resolution of the problem is achieved by the order increase rather than the mesh refinement.

4 Conclusion In this study, analysis and applications of the Spectral Volume Method through applications were presented. This high-order reconstruction is particularly interesting owing to its compact support. The use of flux evaluation on split control volumes makes this method very similar to Finite-Volume method which makes the SVM implementation in existing codes relatively easy. The results are up to expectations with a significant increase in accuracy and a reduction in CPU time compared to a MUSCL method with the same number of elements. The CPU gains are due to the absence of gradient evaluation for inviscid fluxes and the continuity of status variables through internal faces. This property reduces the limitation problems and several cases can be computed with no limitation procedure. For all previously reported assets, the Spectral Volume Method arises as a promising high-order reconstruction mode for both academic and industrial studies especially for complex applications which require great accuracy with computational resource savings.

References 1. Abeele, K., Lacor, C.: An accuracy and stability study of the 2d spectral volume method. J. Comput. Phys. 226(2); 1007–10026 (May 2007) 2. Abeele, K., Ghorbaniasl, M., Parsani, M., Lacor, C.: A stability analysis for the spectral volume method on tetrahedral grids. J. Comput. Phys. 228, 257–265 (2009) 3. Sun, Y., Wang, Z.J., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids vi: Extension to viscous flow. J. Comput. Phys. 215(1), 41–58, (June 2006) 4. Typhon, http://typhon.sourceforge.net/ 5. Wang, Z.J.: Spectral (finite) volume method for conservation laws on unstructured grids. basic formulation: Basic formulation. J. Comput. Phys. 178(1), 210–251 (May 2002) 6. Wang, Z.J., Zhang, L., Liu, Y.: Spectral (finite) volume method for conservation laws on unstructured grids iv: extension to two-dimensional systems. J. Comput. Phys. 194(2), 716–741 (March 2004)

Runge–Kutta Discontinuous Galerkin Method for Multi–phase Compressible Flows Vincent Perrier and Erwin Franquet

Abstract The aim of this contribution is to develop a high order numerical scheme for simulating compressible multiphase flows. For reaching high order, we propose to use the Runge-Kutta Discontinuous Galerkin method. The development of such a method is not straightforward, because it was originally developed for conservative systems, whereas the system of interest is not conservative. We show how to circumvent this difficulty, and prove the accuracy and the robustness of our method on one and two dimensional numerical tests.

1 Introduction Computations of multi-phase flows is known to be a very difficult task. There exist several models, mostly based on the model of [2], which can be written as ∂αk + u I · ∇αk = μ(Pk − Pk¯ ) ∂t ∂ αk ρk + div αk ρk uk = 0 ∂t ∂ α k ρk u k + div αk ρk uk ⊗ uk + ∇(αk Pk ) = PI ∇αk + λ(uk¯ − uk ) ∂t ∂ αk ρk E k + div αk (ρk E k + Pk )uk = PI u I · ∇αk − μPI (Pk − Pk¯ ) ∂t +λu I (uk¯ − uk )

(1)

where αk denotes the volume fraction of each fluid, and ρk , uk , Pk , E k are respectively the density, the velocity, the pressure, and the total energy of the phase k. The total energy, the velocity, and the thermodynamical variables are linked by the V. Perrier (B) INRIA Bordeaux Sud Ouest and Laboratoire de Mathématiques et de leur Applications, Bâtiment IPRA, Université de Pau et des Pays de l’Adour, Avenue de l’Université, 64013 Pau Cedex, France e-mail: [email protected]


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relation E k = εk (Pk , ρk ) + 12 u2k , where εk is a concave function of its arguments. We define k¯ = 1 (resp. = 2) when k = 2 (resp. k = 1). The interfacial velocity and pressure u I and PI must be modeled and this can be a problem, for example for ensuring hyperbolicity of the system. Besides, (1) involves non-conservative products. These ones show two major complications. From a mathematical point of view, they are not defined in case of discontinuities because they are products of distributions. From a numerical point of view, they are not well suited, even with the finite volume method with Godunov fluxes, because they require specific approximations. Last, (1) involves relaxation terms that depend on relaxation parameters λ and μ. They mean that when the two fluids are in contact, they tend to relax to one pressure and one velocity. In this chapter, we concentrate on the simulations in which all the interfaces are solved, i.e. λ = μ = 0. In Sect. 2, we recall how to derive a finite volume scheme for such a system, then we explain in Sect. 3 how to extend this scheme to high order with the Runge-Kutta discontinuous Galerkin (RKDG) method. Last, in Sect. 4, we prove the robustness and accuracy of the scheme with numerical tests.

2 Finite Volume Scheme for Compressible Multiphase Flows The finite volume method on which we are based was developed in [1]. We shortly recall how to derive a numerical scheme for (1) in the case of interface problems. The numerical scheme is based on the derivation of (1) described in [4], for which a probabilist formulation is proposed. We suppose that the space is meshed by conforming cells. At time tn , we know the volume fraction, velocities, and thermodynamic variables of each fluid. The numerical scheme is also based on probabilist ideas: at each realization, each cell is filled with one fluid. The filling of each cell is performed independently. For one realization, the Riemann problem at each side is solved. Then, following Godunov’ method, the solution is projected on a piecewise constant space. For homogeneous Riemann problem, i.e. if the left and right cell of one side are filled with the same fluid, we recover the classical finite volume scheme. If the left and right state are not filled with the same fluid, another flux appears (a Lagrangian flux) which means that one of the fluid is pushing the other fluid inside the cell. Besides, an Eulerian fluid appears too, on the fluid that is present on the boundary of the cell. Last, all the realization are averaged. Finally, the flux on one side is composed of the weighting of the integration of pure phase Riemann problems. This integration includes Eulerian and Lagrangian fluxes. The weights can be computed thanks to the volume fractions. For more details of the numerical scheme, see [1]. We note that for the derivation of the numerical scheme, the system (1) was not used. Actually, only the derivation of (1) made in [4] was used. The continuous system solved by the numerical scheme is therefore not clearly known, but when looking in details in the numerical scheme, we can formally associate the

Runge–Kutta Discontinuous Galerkin Method for Multi–phase Compressible Flows

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average of the Eulerian fluxes to the conservative part of (1), and the averaging of the Lagrangian fluxes to the nonconservative part of (1).

3 Higher Order Extensions For extending the numerical scheme to high order, we propose to use the discontinuous Galerkin method, because this method is compact, so well suited for parallelization, and because it is well suited with unstructured (and possibly hybrid and non-conforming) meshes and with adaptive order. In the discontinuous Galerkin method, the approximation space is a finite elements space in which the test functions are continuous inside the cells of the mesh, but discontinuous on the sides. Thus, a basis of the approximation space is composed of a concatenation of all the basis on a local element. We denote by ϕTi ,k the k th element of the basis in the cell Ti . The discontinuous Galerkin method is based on the projection of the weak formulation of the problem on this approximation space. For example, for the spatial discretization of a steady hyperbolic system divx F(U) = 0, the following equation holds on each cell Ti ∀k

∂ Ti

ϕTi ,k F(U) · n −

Ti

F(U) · ∇ϕTi ,k = 0

(2)

in which the boundary integral is evaluated with a classical finite volume flux [7], and the internal integral is evaluated with a quadrature formula. For time integration, the method of lines is used. The system we are interested in is hyperbolic and nonlinear, so that a stabilization process must be added to the classical discontinuous Galerkin method. Mainly two methods exist: the methods based on addition a streamline diffusion term [6], and the RKDG methods, for which the stabilization is based on slope limiters (see the review [3] and references therein). Here, we propose to use the second method. Indeed, in the multiphase context, we must ensure that the volume fractions remains in [0, 1], and the slope limiting method is the only one which can ensure L ∞ bounds. Applying the RKDG method to a nonlinear hyperbolic problem can be divided into the following steps 1. Evaluate the boundary integrals: This step is straightforward: we use the scheme derived in Sect. 2. 2. Evaluate the internal integrals: This step is not easy, because as pointed out at the end of Sect. 2, the continuous system is not clearly known. 3. Invert the mass matrix: This step is performed as for the classical hyperbolic systems. 4. Slope limiting: As noted in [3], the variables that must be limited are the characteristic variables. Problems 2 and 4 are detailed in what follows.

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3.1 Internal Integral As noted in Sect. 1, the system is not conservative, and this must be taken into account in the internal cell integral. Nevertheless, as noted in Sect. 2, the continuous system is not known, but the nonconservative part can be associated to the averaging of the Lagrangian flux. At each quadrature point of the cell, we know all the variables, but also their gradient. As for the second order extension of [1], we rebuild an interface inside the cell, solve and integrate the Riemann problem, and we weight it by the gradient of the volume fraction. The remaining part of the internal integral is computed as for the conservative case.

3.2 Slope Limiting When applying the RKDG method, a slope limiting must be performed, and this must be done in the characteristic variables [3]. But in our case, the continuous system that is solved is not known as remarked in Sect. 2. In this chapter, we deal only with interface problems: the mixture are only numerical mixtures. This means that either we are at an interface, or we are in a pure fluid. Based on these two cases, we propose to limit as follows: when the flow is a pure fluid, we limit its characteristic variables, and when the flow is a mixture, we limit the volume fraction. In this last case, we keep only the mean value of the other conservative variables. This way of limiting means that the computation remains high order inside pure fluids, and inside advections between pure fluids. In other cases (e.g. shock-interface interaction), the computation is high order for the volume fraction, but not for the other variables inside the diffused interface. For dealing with mixed multiphase flows, this slope limiting shall be improved. Nevertheless, this computation is satisfactory for our numerical tests.

4 Numerical Results All the numerical simulations are made with two fluids. Our aim for this chapter was to compute the Quirk-Karni test [5], so that we will use two fluids, each of them being described by a perfect gas equation of state, Helium (γ = 1.648) and Air (γ = 1.4).

4.1 One Dimensional Shock Tube This first test is the classical Sod shock tube, but in which the left side is filled with Helium, and the right side is filled with Air. Results are shown in Fig. 1.


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1 first order Analytical second order

0,1

0,6 Error

Density

0,8

0,4

0,2

0

First Order Second Order

0

0,2

0,4

0,6

0,8

x

1

0,01

0,001

0,01

0,1

h

Fig. 1 One dimensional multiphase Sod shock tube. On the left, the first and second order solutions of the total density are compared for a mesh of 60 nodes. On the right, we compare the convergence order of the scheme with DG 0 and DG 1 . Of course, we do not recover the convergence order of 1 and 2, because the solution is not regular. Nevertheless, we can see an improvement of the order of the scheme

4.2 Travelling Bubble For this test and the next one, the domain is a rectangle with left bottom corner located on (−0.175, 0) and right top corner on (0.15, 0.0445). A wall boundary condition is imposed on the top and the bottom, and a free stream boundary condition is imposed on the left and on the right. The computational domain is filled by air, with density 1.4 kg m−3 , except for a half bubble of Helium with density 0.25463 kg m−3 . Its center is (0, 0), and its radius is 0.025 m. The mesh for all the two-dimensional tests is an unstructured triangular conforming mesh, composed of 8,932 triangles generated with emc2. In this test, the whole domain has a pressure 1 Pa, and an uniform horizontal velocity of −100 m s−1 . For this test, the exact solution at time 0.001 s is the same bubble, translated of −0.1 m. Numerical results are shown in Fig. 2.

4.3 Shock–Bubble Interaction In this test, the left part of the domain, delimited by x < 0.05 has a pressure of 1 Pa, and is at rest. The right part of the domain has a density of 1.92691 kg m−3 , a velocity of −0.33361 m s−1 , and a pressure of 1.5698 Pa. This condition on the right of the domain induces a shock moving to the left, which will interact with the bubble. Numerical results are shown in Fig. 3.

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Fig. 2 Test of the travelling bubble. Comparison between the first (top) and second (bottom) order solutions: isolines of the volume fraction (α = 0.1; 0.2; ..0.9). We observe that the interface for the second order solution is sharper than for the first order one

Fig. 3 Shock–Bubble Interaction: on the left, the isovalue of the volume fraction at time 0.23. On the right, the density is represented as a greyscale, with 12 isovalues between 0.27 and 2.06, at time 0.0745. We observe a more sharpen interface in the DG 1 case, and also one more density isovalue of the density in the DG 1 case than in the DG 0 case

5 Conclusion In this chapter, we showed how to extend to high order [1] with the discontinuous Galerkin method. It inherits from the main property of this scheme, i.e. it is fully conservative. The main drawback of diffuse interface methods is that the interfaces are very diffused. With this extension to high order, the method exposed could become an opponent to other methods, as the level set methods. Acknowledgements The first contributing author is very grateful to Dr. Caroline Baldassari (INRIA – Total), who emailed him all the numerical results he had forgotten on a USB key in Pau.


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References 1. Abgrall, R., Saurel, R.: Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186(2), 361–396 (2003) 2. Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. Int. J. Multiphase Flows. 12(12), 861–889 (1986) 3. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. J. Sci. Comput. 16(3), 173–261 (2001) 4. Drew, D.A., Passman, S.L.: Theory of multicomponent fluids, vol. 135. Applied Mathematical Sciences. Springer, New York, NY, (1999) 5. Quirk, J.J., Karni, S.: On the dynamics of a shock-bubble interaction. J. Fluid Mech. 318, 129–163 (1996) 6. Szepessy, A.: Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions. Math. Comp. 53(188), 527–545 (1989) 7. Toro, E.F.: Riemann solvers and numerical methods for fluid dynamics 2nd edn. A Practical Introduction. Springer, Berlin (1999)

Energy Stable WENO Schemes of Arbitrary Order Nail K. Yamaleev and M.H. Carpenter

Abstract A systematic approach for constructing Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference schemes of arbitrary order is presented. The new class of schemes is proven to be stable in the energy norm for both continuous and discontinuous solutions of systems of linear hyperbolic equations. We also present new weight functions and constraints on their parameters, which provide consistency and much faster convergence of the high-order ESWENO schemes to their underlying linear schemes. Furthermore, the improved weight functions guarantee that the ESWENO schemes are design-order accurate for smooth solutions with arbitrary number of vanishing derivatives and provide much better resolution near strong discontinuities than the conventional counterparts. Numerical results show that the new ESWENO schemes are stable and significantly outperform the corresponding WENO schemes of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities.

1 Energy Estimate Consider a linear, scalar wave equation ∂u + ∂∂ xf = 0, f = au, ∂t u(0, x) = u 0 (x)

t ≥ 0,

0 ≤ x ≤ 1,

(1)

where a is a constant, and u 0 (x) is a bounded piecewise continuous function. Without loss of generality, assume that a ≥ 0 and the problem is periodic on 0 ≤ x ≤ 1.

N.K. Yamaleev (B) North Carolina A&T State University, Greensboro, NC, USA e-mail: [email protected]


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Equation (1) is discretized as follows: ∂u + P −1 Q f = − P −1 D1 n Λn [D1 n ]T f, ∂t N

(2)

n=1

where f = au, u = [u 0 (t), u 1 (t), . . . , u J (t), ]T is the discrete approximation of the solution u of Eq. (1), and D1 is a first-order backward difference operator. The matrices P, Q, and Λ are nonlinear and satisfy the following summation-by-parts (SBP) convention: N ∂ ¯f D1 n Λn [D1 n ]T = D ¯f + O(Δx p ) with D = P −1 [Q + R], R = ∂x n=1

Q+

QT

= 0;

R=

RT ,

vT

Rv ≥ 0;

P=

PT ,

vT

(3)

Pv > 0, ∀v = 0.

To show that the above finite difference scheme is stable, the energy method is used. Multiplying Eq. (2) with uT P, we have N

T 1 d 2 T [D1 n ]T u Λn [D1 n ]T u, u P + au Qu = −a 2 dt

(4)

n=1

where · P is the P norm (i.e., u2P = uT Pu). Adding Eq. (4) to its transpose and accounting for the periodic boundary conditions and the skew-symmetry of Q yield N

T d [D1 n ]T u Λn [D1 n ]T u ≤ 0 . u2P = −2a dt

(5)

n=1

The right-hand side of Eq. (5) is nonpositive because the diagonal matrices Λn , n = 1, N are positive semidefinite (i.e., vT Λn v ≥ 0 for all real v of length (J + 1)) and a ≥ 0; thus the stability of the finite difference scheme given by Eq. (2) is assured. This result can be summarized in the following theorem: Theorem 1 The approximation (2) of the problem (1) is stable if Eq. (3) holds. Remark 1 As will be shown in Sect. 3, conventional WENO schemes can be recast in the form of Eq. (2). Note, however, that the diagonal matrices Λn in the WENO dissipation operator are not positive semidefinite, thus no energy estimate is available for the conventional WENO schemes.

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2 Conventional WENO Schemes with New Weight Functions We now present a one-parameter family of fifth-order WENO schemes and then use it as a starting point in the development of Energy Stable WENO schemes. The methodology is directly applicable to WENO schemes of arbitrary order; ESWENO schemes up to eighth order are presented in [3]. Any conventional high-order WENO finite difference scheme for the scalar onedimensional wave equation (1) can be written in the following semidiscrete form: fˆj+ 1 − fˆj− 1 du j 2 2 + = 0. dt Δx

(6)

For the fifth-order WENO scheme developed in [1], the numerical flux fˆj+ 1 2 is computed as a convex combination of three third-order fluxes defined on the following stencils: SL L = {x j−2 , x j−1 , x j }, SL = {x j−1 , x j , x j+1 }, and S R = {x j , x j+1 , x j+2

}. Note that this set of stencils is not symmetric with

respect to the j + 12 point; thus, the fifth-order WENO scheme is biased in the upwind direction. A central WENO scheme can be constructed from the fifth-order WENO scheme by including an additional downwind candidate stencil S R R = {x j+1 , x j+2 , x j+3 }, which is given by L LL L R R RR , fˆj+ 1 = w Lj+1/2 f j+1/2 + w Lj+1/2 f j+1/2 + w Rj+1/2 f j+1/2 + w Rj+1/2 f j+1/2 2

(7)

(r ) where f j+1/2 , r = {L L , L , R, R R} are third-order fluxes defined on these four stencils. In general, high-order conventional central WENO schemes that are built in this manner are unstable when unresolved features are present in the domain. The terms w L L, w L , w R , and w R R in Eq. (7) are nonlinear weight functions. We now present new weight functions that provide faster convergence of the WENO schemes to the corresponding underlying linear schemes for smooth solutions, and deliver improved shock-capturing capabilities near unresolved features. For the fifth-order WENO schemes, these weights and smoothness indicators are given by

w(r ) 1 = j+ 2

αr , αl

αr = d (r ) 1 +

τp ε+βr

, r = {L L , L , R, R R},

(8)

l

where ε is a small positive parameter that depends on Δx. The functions β (r ) are the classical smoothness indicators (see [1] for the general expression). The preferred values d (r ) are given by dLL =

1 10

− ϕ ; dL =

6 10

− 3ϕ ; d R =

3 10

+ 3ϕ ; d R R = ϕ ,

(9)

where ϕ is a parameter. The convergence rate of the scheme (6–9) with w(r ) = 1 for which the d (r ) , r = {L L , L , R, R R} is equal to 5 for all ϕ except ϕc = 20

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convergence rate is 6. The classical fifth-order upwind-biased WENO scheme of Jiang and Shu is obtained for ϕ = 0. The expression for τ5 is defined as τ5 =

+ 5 f j−1 − 10 f j + 10 f j+1 − 5 f j+2 + f j+3 −f 2 j−2 f j−2 − 4 f j−1 + 6 f j − 4 f j+1 + f j+2 ,

2

, for ϕ = 0 (10) for ϕ = 0,

which represents a quadratic function of the fifth-degree undivided difference defined on the entire six-point stencil and provides the highest order information that is possible on this stencil. A detailed consistency analysis and sufficient conditions on the parameters of the new weight functions that guarantee the design order of accuracy for both the WENO and ESWENO schemes are presented in [2, 3].

3 Energy Stable WENO Schemes A systematic approach is needed to construct “energy stable” modifications for existing WENO schemes of any order. The following algorithm transforms an existing WENO scheme into an ESWENO scheme that is stable in an energy norm. First, the conventional fifth-order WENO scheme (6) with the flux fˆj+ 1 given by Eq. (7) 2 is represented in a matrix form. An explicit expression for the differentiation matrix D 5 is given in [3]. The derivative matrix D 5 is then decomposed into symmetric 5 5 , where D 5 and skew-symmetric parts D 5 = Dskew + Dsym skew and the norm take 5 −1 T 5 is the form: Dskew = P Q 5 ; Q 5 + Q 5 = 0; P = Δx I, while the matrix Dsym expressed as 5 Dsym = P −1

T T T T + D12 Λ52 D12 + D11 Λ51 D11 + D10 Λ50 D10 D13 Λ53 D13 . (11)

The matrices Λ5k , k = 0, 3 in Eq. (11) are diagonal with the jth element given by L R w Lj+5/2 − w Rj+1/2 1 L L R R w Lj+3/2 = 12 − 4w Lj+5/2 + w Lj+3/2 − w Rj+1/2 + 4w Rj−1/2 − w Rj+1/2 1 L L L 3w Lj+1/2 = 12 − 5w Lj+3/2 + 2w Lj+5/2 + w Lj+1/2 − w Lj+3/2 R R R +w Rj−1/2 − w Rj+1/2 − 2w Rj−3/2 + 5w Rj−1/2 − 3w Rj+1/2 L R = 12 −w Lj−1/2 − w Lj−1/2 − w Rj−1/2 − w Rj−1/2 L R =0. +w Lj+1/2 + w Lj+1/2 + w Rj+1/2 + w Rj+1/2 (12)

5 λ3 j, j = 5 λ2 j, j 5 λ1 j, j

5 λ0 j, j

1 6


85

(r ) Because = 1, Λ50 is always equal to zero, and is not included in r w: 5 . The diagonal terms λ 5 Dsym i j, j , i = 1, 2, 3 could be either positive or negative; thus, the conventional fifthand sixth-order WENO schemes may become locally unstable. Defining λ¯ i5 j, j , i = 1, 2, 3 to be smoothly positive λ¯ i5 j, j = 5 2 1 2 − λ5 ¯5 + δ λ i j, j i j, j , the additional dissipation operator becomes Dad = i 2 3 i 5 i T D1 Λ¯ i D1 , and the resulting energy stable scheme is obtained P −1 i=1 by adding the additional dissipation term to the original WENO scheme. That 5 . By construction, D¯ 5 satisfies all of the conditions of is, D¯ 5 = D 5 + D¯ ad Theorem 1, thereby providing stability of the fifth- and sixth-order ESWENO 5 with the new weight funcschemes. Note that the additional dissipation term D¯ ad tions is design-order accurate (see [3] for details).

4 Numerical Results We now assess the performance of the fifth-, and sixth-order ESWENO schemes with the new weights and compare them with the conventional WENO counterparts. As has been proven in Sect. 3, all eigenvalues of the symmetric part of the ESWENO operator are nonpositive. In contradistinction to the ESWENO scheme, the symmetric part of the WENO operator may have positive eigenvalues, thus indicating that the conventional WENO scheme may become locally unstable. These properties of the WENO and ESWENO schemes are shown in Fig. 1. 5th-order WENO 5th-order ESWENO

101

max[Re(eigenvalue)]

10–1 10–3 10–5 10–7 10–9 10–11 10–13 10–15

2

4

Time

6

8

10

Fig. 1 Time histories of the rightmost eigenvalue of the symmetric part of fifth-order WENO and ESWENO operators for the Gaussian pulse problem

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Log10 || Error ||max

–3 –4 –5 –6

–1

–7

–3 –4 –5 –6 –7 –8 –9

–8 1.5

6th-order linear 6th-order WENO 6th-order ESWENO

–2 Log10 || Error ||max

5th-order linear 5th-order WENO 5th-order ESWENO

–2

–10 2

2.5 Log10(N)

3

1.5

3.5

2

2.5 Log10(N)

3

3.5

Fig. 2 L ∞ error norms obtained with the fifth- (left) and sixth-order (right) linear, WENO, and ESWENO schemes for the Gaussian pulse propagation problem

The L ∞ error norms obtained with the fifth- and sixth-order WENO and 2 ESWENO schemes for a linear convection problem with u(0, x) = e−300(x−0.5) are depicted in Fig. 2. Both the fifth- and sixth-order ESWENO schemes are equal in accuracy to the corresponding underlying linear schemes, as seen in Fig. 2. The L ∞ error norm of the fifth-order WENO scheme is nearly an order of magnitude larger than that of the corresponding fifth-order ESWENO scheme. In contrast to the sixth-order ESWENO scheme, the conventional sixth-order WENO scheme shows

exact 5th-order WENO 5th-order ESWENO 4.5 4 3.5

Density

3 2.5 2 1.5 1 0.5

–4

–3

–2

–1

0 x

1

2

3

4

Fig. 3 Density profiles computed with the fifth-order WENO and ESWENO schemes on uniform grid with 301 grid points for the shock density wave interaction problem


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only fifth-order convergence and is quite far from the theoretical limit represented by the corresponding sixth-order central linear scheme. The last problem is the interaction of a shock and an entropy wave (see [1] for details). The exact solution to this problem is not available. Therefore, a numerical solution obtained with the conventional fifth-order WENO scheme on a uniform grid with J = 4,000 grid cells is used as a reference solution. Numerical solutions computed with the fifth-order WENO and ESWENO schemes on a 301-point grid at t = 1.8 are compared with the “exact” reference solution in Fig. 3. All the WENO and ESWENO solutions are free of spurious oscillations. Note, however, the ESWENO solution is much less dissipative than that obtained with the conventional WENO counterpart.

References 1. Jiang, G., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 202–228 (1996) 2. Yamaleev, N.K., and Carpenter, M.H.: Third-order energy stable WENO scheme. J. Comput. Phys. 228(8), 3025–3047 (2009) 3. Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order Energy Stable WENO Schemes. J. Comput. Phys. 228(11), 4248–4272 (2009)

Part IV

Two-Phase Flow

A Hybrid Method for Two-Phase Flow Simulations Kateryna Dorogan, Jean-Marc Hérard, and Jean-Pierre Minier

Abstract This chapter proposes a new hybrid method for modelling and computing turbulent two-phase flows. It consists in coupling Eulerian and Lagrangian formulations for description of the dispersed phase while the properties of the fluid remain calculated with the Eulerian approach. For that purpose we introduce a relaxation model in order to stabilize approximations of the Eulerian part of the particle phase. We present an analytic validation and numerical results when coupling with the Lagrangian data.

1 Introduction and Governing Equations Poly-dispersed two-phase flows consisting of a turbulent carrier phase (a liquid or a gas) and dispersed particles (solid particles, droplets or bubbles) are complex processes which are very important in many industrial situations. Therefore, an accurate prediction of these flows is required for engineering purposes. Furthermore, for the modelling and numerical simulation of a poly-dispersed two-phase flow, the two phases (fluid and particles phase) have to be treated in a coupled way. However, the two most popular approaches (Eulerian and Lagrangian) for two-phase flow modelling have disadvantages which limit their capacities. Hybrid methods try to gather the advantages of the Eulerian approach (expected values free from statistical error and low calculation costs) with those of the PDF-approach (polydispersity and non-linear local source terms are treated without approximations) [9, 10]. At the moment, most hybrid methods use only one description (Eulerian or Lagrangian) for each phase. This chapter is aimed at proposing a new hybrid method for modelling and computing turbulent two-phase flows. The idea of this hybrid method is to use a mixed Eulerian/Lagrangian formulation for the dispersed phase while the properties of the fluid remain calculated with the Eulerian approach [3]. By treating some statistical quantities (such as the mean particle velocity and the local particle concentration) in K. Dorogan (B) EDF R&D, MFEE, F-78400 Chatou, France; LATP, CMI, 13453 Marseille, France e-mail: [email protected]


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an Eulerian manner (i.e. as continuous variables on a fixed grid), the overall system will be less sensitive to errors introduced by statistical noise. This, in turn, will allow for computations using a smaller number of computational particles, thus leading to faster calculations. Therefore, in this new hybrid method, the two approaches are coupled in a fully consistent manner while keeping the high level of physical information provided by the Lagrangian description. In the Lagrangian part, the particle description treats the convection term and right-hand-side terms without approximation. The models are particle stochastic models for which specific numerical schemes are used: d x p,i = U p,i dt dU p,i =

1 τ p (Us,i

dUs,i = − ρ1f

− U p,i )dt

∂P ∂ xi dt

+

∂ U f,i (Us,i −U f,i ) dt+ E U p, E∗ j − U f, j ∂ x j dt −

(1)

TL ,i

˜ + 2 (bi k/k ˜ − 1) dWi , + ε C0 bi k/k 3 where x p (t) is the particle position, U p (x p (t), t) – the particle velocity, U f (x f (t), t) – the fluid velocity and Us (x p (t), t) – the fluid velocity seen (fluid velocity sampled along the particle trajectory x p (t)) [9]. In practice, these methods appear as particle/mesh Monte Carlo methods [11]. Here, to guarantee the coupling of methods for the particle description we have to choose the averaging and interpolation procedures, that is how information is exchanged from the mesh to the stochastic particles (interpolation of Eulerian mean fields at particle positions) and from the stochastic particles to the mesh (how mean fields are extracted). Moreover, such a coupling introduces noisy quantities (computed by the stochastic equations) in the Eulerian part of the model. This poses a problem since the Eulerian description of the particle motion is given by a system of equations which presents an important convective part and thus requires a stabilization. This corresponds to the left-hand side of: ∂α pE ρ p ∂t

+

E ∂ α pE ρ p U p,i

E ∂α pE ρ p U p,i ∂t

∂ xi

+

=0

E U E +u u L ∂ α pE ρ p U p,i p,i p, j p, j ∂x j

=

α pE ρ p g

+

α pE ρ p Ur,i τp

(2)

In order to tackle this problem, a specific relaxation procedure is proposed. Then the system of partial differential equations is simulated by a Finite Volume Method relying both on up-winding techniques and relaxation tools.

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2 Basic Approach

In order to compute stable approximations of the variable W t = α pE , α pE U pE , we proceed as follows. We first introduce a new variable Z t = α pE , α pE U pE ,

E α pE R p,i j , and a relaxation system of the form: ∂α pE ρ p ∂t

+

E ∂α pE ρ p U p,i

E ∂α pE ρ p U p,i ∂t E ∂α pE ρ p R p,i j ∂t

∂ xi

+ +

=0

E E ∂α pE ρ p U p, j U p,i ∂x j ∂ρ p Fi j (Z ) ∂x j

+

E ∂α pE ρ p R p,i j ∂x j

= α pE ρ p g +

α pE ρ p Ur,i τp

(3)

E + ρ p Ai jk (Z ) ∂∂xZk = α pE ρ p u p,i u p, j L − R p,i j /τ R

Two distinct evolution operators, or alternatively two tensorial forms for the couple (Fi j (Z ), Ai jk (Z )) may be introduced. The first one, refered to as (A1), relies on the pure Eulerian formulation and thus takes advantage of the hyperbolic structure of Eulerian closures [1]. This was initiated in [6] and then extended in [7]. Approximations of solutions are obtained using the approximate Godunov scheme introduced in [2]. A nice feature in this approach is that the whole set of partial differential equations in the evolution step preserves the realisability of the “Reynolds stress E , both at the continuous and discrete levels. This is in fact mandatory tensor” R p,i j E n remains since eigenvalues remain real if and only if the quadratic form n i R p,i j j positive. However, a drawback in this approach is due to the non-conservative form of the governing equations in the evolution step corresponding to the left hand-side of (3). Thus, non-conservative products that are effective in genuinely non-linear fields are not uniquely defined. A straightforward consequence is that multiple shock solutions might arise. This has motivated the introduction of a second couple (Fi j (Z ), Ai jk (Z )) – corresponding to (A2) (see [4]). The main objective here is to comply with the same specifications: (i) the system should be hyperbolic, (ii) E n , (iii) jump conthe Reynolds stress tensor should remain realisable 0 ≤ n i R p,i j j ditions should be uniquely defined, field by field. The basic idea is to introduce a couple (Fi j (Z ), Ai jk (Z )), which is close enough to the first one, but such that non-conservative products are only effective through linearly degenerated fields [8]. An evolution step, computed with a fractional step method, is followed by a E L relaxation procedure which enables to restore local values of R p,i j = R p,i j at the beginning of the next time step. In a 1D framework, setting ρ = α pE ρ p and U = E , the evolution step for R = R E U p,1 p,11 in the approach (A2) reads: ∂(ρ R) a02 ∂U ∂(ρ R) +U + = 0, ∂t ∂x ρ ∂x

(4)

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whereas in the approach (A1) it reads: ∂(ρ R) ∂U ∂(ρ R) +U + 3ρ R = 0. ∂t ∂x ∂x

(5)

The multi-dimensional extension of (A2) is detailed in [4]. An exact Godunov solver is used to compute the evolution step [5], and the instantaneous relaxation step reads: ρ n+1 = (ρ ∗ )n+1 ,

U n+1 = (U ∗ )n+1 ,

(ρ R)n+1 = (ρ ∗ )n+1 (R Lag )n+1 .

(6)

3 Numerical Results 3.1 Analytical Test Cases In order to validate the two approaches we consider some test cases where analytical solutions are known and we focus especially on the most difficult configurations. We assume the following closure relation: ρ R = S0 ρ γ with constant entropy 3 Approach 1, N = 100 Approach 1, N = 1000 Approach 2, N = 100 Approach 2, N = 1000 exact solution

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–0,4

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Fig. 1 Symmetric double shock: comparison of density and velocity approximations


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ln(error_velocity)

ln(error_density)

S0 = 105 and γ = 3. This value of γ corresponds to the isentropic case arising in [1, 7]. We focus on two 1D Riemann problems herein; the computational domain is [−0.5, 0.5], we use a time step such that CFL = 1/2, and the regular meshes contain from 100 up to 105 cells. We also plot the L 1 -norm of the error wrt the mesh size. Symmetrical double shock wave: This case is interesting in order to compare the stability of both schemes when strong shock waves develop. We take the following initial conditions: ρ L = ρ R = 1 and U L = −U R = 1, 000. Approximations with both methods converge towards the correct solution (Fig. 1) with an asymptotic rate of convergence around 1. Moreover, the accuracy for the two schemes is almost the same for a given mesh size (Fig. 2). Eventually, we emphasize that the relaxation approach (A2) exhibits a better stability since (A1) induces tiny – stable – oscillations near the shocks. An important ingredient that contributes to the success of (A1) is that the conservative form of equations (2) is preserved. 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15 –13 –2 –3 –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15 –13

Approach1 Approach2

–12

–11

–10

–9 –8 ln(h)

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–7

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Fig. 2 L1 convergence curves for symmetric double shock: density (left), velocity (right). Coarser mesh: 100 cells; finer mesh: 100,000 cells

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–4

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–5

–6

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–8 ln(h)

Fig. 3 L 1 convergence curves for symmetric double rarefaction with a vacuum: density (left), flow rate (right). Coarser mesh: 100 cells; finer mesh: 100,000 cells

Symmetrical double rarefaction wave with vacuum occurence: We now examine a case with initial conditions: ρ L = ρ R = 1, U L = −U R = −1, 000. The exact solution is composed of two symmetric rarefaction waves. The intermediate state exhibits a vacuum zone where the velocity is not defined but the flow rate and the density are null. Both approximations of (ρ, Q) obtained with (A1) and (A2) converge towards the correct solution (see Fig. 3) with a numerical rate close to 0.7.

3.2 Numerical Results with Noisy Reynolds Stresses Here, noisy Reynolds stresses are plugged in the Eulerian system of equations at each time step in the cells that belong to the region x ∈ [−0.25, 0.25] as follows: γ (ρ R)in = S0 ρin (1 + r ms(0.5 − rand(0, 1))),


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2 rms = 0.5, Approach 1 rms = 0.5, Approach 2 rms = 0, Approaches 1,2

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velocity

100 0 –100 –200 –300

–0,4

–0,2

0 x

0,2

0,4

Fig. 4 Subsonic shock tube: density and velocity approximations using (A1) and (A2)

where r ms stands for the noise intensity and rand allows to manage the noise amplitude. We choose the initial conditions of a subsonic shock tube problem: ρ L = 1, ρ R = 0.24 and U L = −10, U R = −282. We note that the noisy region is not developping in time and only some propagated waves disturb the smooth approximations outside of the noisy domain, Fig. 4. A meansquare linear approximation of the L 1 difference curves (between solutions with r ms = 0.5 and r ms = 0) exhibits a slope that is very close to 0; this confirms the fact that the noise does not increase with time. The same remark holds for other values of the noise intensity. Other test cases with noisy data [4] have shown that the noise is not diminishing when the mesh is refined. Moreover, the L 1 norm of the difference between the approximations with r ms = 0.1 and r ms = 0 tends to be constant when h vanishes. Eventually, we have noticed that the difference between noisy approximations and those without a noise is increasing with increasing r ms in a linear manner. Acknowledgements Part of the financial support of the first author has been provided by ANRT (Association Nationale de la Recherche Technique, Ministère de la Recherche) through a EDF/CIFRE contract, and also by FSE (Fonds Social Européen).

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References 1. Brun, G., Hérard, J.M., Jeandel, D., Uhlmann, M.: An approximate Riemann solver for a class of realizable second order closures. Int. J. Comp. Fluid Dyn. 13(3), 223–249 (1999) 2. Buffard, T., Gallouët, T., Hérard, J.M.: A sequel to a rough Godunov scheme. Application to real gases. Comput. Fluids. 43, 813–847 (2000) 3. Chibbaro, S., Hérard, J.M., Minier, J.P.: A novel Hybrid Moments/Moments-PDF method for turbulent two-phase flows. Final Technical Report Activity Marie Curie Project. TOK project LANGE Contract MTKD-CT-2004 509849 (2006) 4. Dorogan, K., Hérard, J.M., Minier, J.P.: Development of a new hybrid method for gas-particle two-phase flow modelling. Internal EDF report (2010) 5. Godlewski, E., Raviart, P.A.: Numerical approximation of hyperbolic systems of conservation laws. AMS, 118, Springer, New York, NY (1996) 6. Hérard, J.M., Minier, J.P., Chibbaro, S.: A Finite Volume scheme for hybrid turbulent twophase flow models. AIAA paper 2007-4587, http://aiaa.org (2007) 7. Hérard, J.M., Uhlmann, M., Van der Velden, D.: Numerical techniques for solving hybrid Eulerian Lagrangian models for particulate flows. EDF report H-I81-2009-3961-EN (2009) 8. Jin, S., Xin, Z.: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48, 235–276 (1995) 9. Minier, J.P., Peirano, E.: The pdf approach to polydispersed turbulent two-phase flows. Phys. Rep. 352, 1–214 (2001) 10. Muradoglu, M., Jenny, P., Pope, S.B., Caughey, D.A.: A consistent hybrid finite-volume/ particle method for the pdf equations of turbulent reactive flows. J. Comp. Phys. 154, 342–371 (1999) 11. Peirano, E., Chibbaro, S., Pozorski, J., Minier, J.P.: Mean-field/PDF numerical approach for polydispersed turbulent two-phase flows. Prog. Ene. Comb. Sci. 32, 315–371 (2006)

HLLC-Type Riemann Solver for the Baer–Nunziato Equations of Compressible Two-Phase Flow Svetlana A. Tokareva and Eleuterio F. Toro

Abstract A complete, fully nonlinear approximate Riemann solver of the HLLC type for the Baer–Nunziato equations of compressible two-phase flow is constructed. The new HLLC solver is assessed in the framework of finite volume, discontinuous Galerkin finite element and path-conservative schemes.

1 Introduction The Baer–Nunziato equations are a three-dimensional time-dependent system of eleven equations with source terms that model the dynamics of a flowing mixture of two compressible materials or phases, typically a solid particle phase and a gaseous phase. The model was first proposed in [1] in the context of granular energetic combustible materials embedded in gaseous combustion products, see also [6, 8]. The equations of the model are hyperbolic, except for some well identified situations, and the complete mathematical structure of the 1D system is available [3]. However, the equations cannot be written in conservation-law form, or divergence form. This chapter is primarily concerned with the Riemann problem for the Baer–Nunziato equations. To construct the Riemann solver, we first apply the HLLC approach based on integral averaged Rankine-Hugoniot relations across the non-linear waves, for each phase. We then connect the obtained states across the solid linearly degenerate field using jump conditions based on a thin-layer theory reported in [7]. The resulting HLLC-type approximate Riemann solver is fully non-linear and complete.

S.A. Tokareva (B) Seminar for Applied Mathematics, ETH Zurich, 8092 Zurich, Switzerland e-mail: [email protected]


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2 Equations and the Riemann Problem 2.1 One-Dimensional Baer–Nunziato Equations The homogeneous one-dimensional Baer–Nunziato equations are a set of seven PDEs: ∂t Q + ∂x F(Q) + T(Q)∂x α¯ = 0,

(1)

where ¯ αρ, αρu, αρ E]T ; Q = [α, ¯ α¯ ρ, ¯ α¯ ρ¯ u, ¯ α¯ ρ¯ E,

F = [0, α¯ ρ¯ u, ¯ α¯ ρ¯ u¯ 2 + p¯ , α¯ u¯ ρ¯ E¯ + p¯ , αρu, α ρu 2 + p , αu (ρ E + p)]T ; T = [u, ¯ 0, − p, − p u, ¯ 0, p, p u] ¯ T. Here ρ, u, p, E are gas density, velocity, pressure and total energy, and ρ, ¯ u, ¯ p, ¯ E¯ are the corresponding variables for the solid; α and α¯ are volume fractions. We assume an ideal equation of state (EOS) for the gas phase and a stiffened EOS for the solid phase: p = (γ − 1)ρe, p¯ = (γ¯ − 1)ρ¯ e¯ − γ¯ P¯0 , where e and e¯ are the specific internal energies, γ and γ¯ are the specific heat ratios of the gas and solid phases, respectively, and P¯0 is a known constant. Solid and gas volume fractions are related through the saturation condition: α¯ + α = 1.

2.2 Exact Solution of the Riemann Problem Consider the Riemann problem for the homogeneous Baer–Nunziato equations, namely ∂t Q + ∂x F(Q) + T(Q)∂x α¯ = 0, Q L , if x < 0; Q(x, 0) = Q R , if x > 0.

(2) (3)

The structure of the exact solution is illustrated in Fig. 1. There are in general six distinct wave families: three for the gas phase and three for the solid phase. These six waves separate seven constant states. For the solid phase there are two intermediate regions of constant states “∗ L” and ∗ “ R” separated by the solid contact, in which densities and pressures are different and velocities are equal: u¯ ∗L = u¯ ∗R . For the gas phase the intermediate values are presented by three constant regions, denoted by “∗ L”, “∗ 0” and “∗ R”. The exact values for these states are in general dependent on the relative positions of the two contact waves. For the configuration shown in Fig. 1 we have the following ∗ 1/γ p definitions of the variables in region “∗ 0”: u ∗0 = u ∗R , p0∗ = p ∗R , ρ0∗ = ρ L∗ p∗R . L

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Fig. 1 Intermediate states for solid (left) and gas (right) phases

Expressions for ρ¯ L∗ , u¯ ∗L , p¯ ∗L , ρ¯ ∗R , u¯ ∗R , p¯ ∗R and ρ L∗ , u ∗L , p∗L , ρ ∗R , u ∗R , p∗R can be obtained by considering the Rankine-Hugoniot conditions across the appropriate non-linear wave [9]. An important feature of the Baer–Nunziato equations is the fact that the jump of the solid phase volume fraction occurs only across the solid contact discontinuity, which means that the two phases remain decoupled away from the solid contact, and Eq. (1) reduce to a pair of Euler equations for each phase separately.

3 An HLLC-type Riemann Solver We follow the basic ideas of [7] in construction of our approximate Riemann solver. In this chapter we deal only with the “subsonic” case, when the solid contact is situated between the left and right gas-phase waves on the wave pattern in x − t plane. For one of the two possible “subsonic” wave configurations (Fig. 1) the thin layer equations take the following form: u¯ ∗R − u¯ ∗L = 0; ∗ 1/γ ∗ pR αR u R − u¯ ∗R − α L u ∗L − u¯ ∗L = 0; ∗ pL ∗ α¯ R p¯ R + α R p∗R − α¯ L p¯ ∗L − α L p ∗L + α L ρ L∗ u ∗L − u¯ ∗L u ∗R − u ∗L = 0; ∗ 1/γ 2 γ p ∗R pL γ p∗L 1 ∗ 1 ∗ ∗ 2 u u L − u¯ ∗L = 0; + − u ¯ − − R R ∗ ∗ ∗ (γ − 1)ρ L p R 2 (γ − 1)ρ L 2 ∗ 1/γ pR , u ∗0 = u ∗R , p0∗ = p ∗R . ρ0∗ = ρ L∗ p ∗L We combine the HLLC approach with the numerical solution of thin layer equations across the solid contact. We treat the left and right waves of each phase as discontinuities propagating with known velocities S¯ L , S¯ R , SL and S R , respectively. There is no change of phase volume fractions across these waves, therefore the

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two-phase equations reduce to a pair of Euler equations. From the averaged Rankine-Hugoniot conditions we can obtain direct expressions for the intermediate states, which are absolutely the same as the HLLC relations for pure fluids, see [9]:

S¯ L − u¯ L S¯ R − u¯ R ∗ , ρ¯ R = ρ¯ R , = ρ¯ L S¯ L − u¯ ∗L S¯ R − u¯ ∗R p¯ ∗L = p¯ L + ρ¯ L ( S¯ L − u¯ L ) u¯ ∗L − u¯ L , p¯ ∗R = p¯ R + ρ¯ R ( S¯ R − u¯ R ) u¯ ∗R − u¯ R . ρ¯ L∗

Analogous expressions are valid for “∗ L” and “∗ R” states in the gas phase. We use Newton’s method to solve the resulting nonlinear system.

4 Computational Results We have tested the performance of our Riemann solver in three numerical methods: finite-volume (denoted as FV) [7], Runge-Kutta discontinuous Galerkin (DG) [2, 5] and path-conservative (PC) [4]. Figures 2 and 3 present the results for the gas and solid densities and velocities computed using the mesh of 100 cells in the

exact FV DG PC

0.8 0.7

exact FV DG PC

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1 0.3 0

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exact FV DG PC

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Fig. 2 Results for the gas phase: computed (symbol) and exact solution (line) at time t = 0.007

1

HLLC-Type Riemann Solver for the Baer–Nunziato Equations 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

exact FV DG PC

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Fig. 3 Results for the solid phase: computed (symbol) and exact solution (line) at time t = 0.007

one-dimensional domain [0, 1] at Courant number equal to 0.9, except for the RKDG method, for which the Courant number was 0.6. Computational results show that the developed Riemann solver produces accurate intermediate states and can be implemented in finite-volume, path-conservative and discontinuous Galerkin schemes. The solution obtained by finite-volume scheme with HLLC-type solver is equivalent to the one computed with the exact solver. However, utilisation of the derived Riemann solver in numerical schemes saves up to 50% of computaional time compared to the exact solver.

5 Concluding Remarks We have computed approximate solutions of the Riemann problem for the split three-dimensional Baer–Nunziato equations of compressible two-phase flow for ideal and stiffened gas equations of state. The Riemann solver is non-linear and complete, as it contains all the characteristic fields present in the exact solution of the Riemann problem. The main use of the proposed Riemann solver is in the calculation of numerical fluxes for numerical methods intended for solving the general initial-boundary value problem.

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The approximate Riemann solver of this chapter can be used in a straightforward manner to solve the two- and three-dimensional Baer–Nunziato equations. The solver could also be extended to deal with more complicated equations of state in order to solve realistic problems. Acknowledgements The first author acknowledges the financial support of the Italian ministry of research and education (MIUR) in the framework of the research project PRIN2007, that in part made possible the work of this article.

References 1. Baer, M.R., Nunziato, J.W.: A Two-Phase Mixture Theory for the Deflagration-to-Detonation Transition (DDT) in Reactive Granular Materials. J. Multiphase Flow 12, 861–889 (1986) 2. Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convectiondominated problems. J. Sci. Comp. 3, 173–261 (2001) 3. Embid, P., Baer, M.: Mathematical analysis of a two-phase continuum mixture theory. Continuum Mech. Thermodyn. 4, 279–312 (1992) 4. Parés, C.: Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J. Numer. Anal. 44, 300–321 (2006) 5. Rhebergen, S., Bokhove, O., van der Vegt, J.J.W.: Discontinuous Galerkin Finite Element Methods for Hyperbolic Nonconservative Partial Differential Equations. J. Comput. Phys. 227, 1887–1922 (2008) 6. Saurel, R., Abgrall, R.: A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows. J. Comp. Phys. 150, 425–467 (1999) 7. Schwendeman, D.W., Wahle, C.W., Kapila, A.K.: The Riemann Problem and a High-Resolution Godunov Method for a Model of Compressible Two-Phase Flow. J. Comp. Phys. 212, 490– 526 (2006) 8. Stewart, B., Wendroff, B.: Two-Phase Flow: Models and Methods. J. Comp. Phys. 56, 363–409 (1984) 9. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag Berlin Heidelberg (2009)

Parallel Direct Simulation Monte Carlo of Two-Phase Gas-Droplet Laser Plume Expansion from the Bottom of a Cylindrical Cavity into an Ambient Gas Alexey N. Volkov and Gerard M. O’Connor

Abstract A combined computational model for simulation of the expansion of twophase laser plumes is developed. The model includes a two-dimensional thermal model of the irradiated target, Direct Simulation Monte Carlo method for flow of multi-component gas mixture in the plume, and a Lagrangian scheme for tracking of trajectories of individual sub-micron droplets generated on the irradiated surface. The model is implemented in a parallel computational code and applied for simulations of the plume expansion into an ambient gas, which is induced by a nanosecond laser pulse irradiating the bottom of a cylindrical cavity on the target surface. Simulations reveal the significant physical effects of the ambient gas chemical composition on the motion of laser ablated submicron debris in the vicinity of the target.

1 Introduction The efficiency of deep drilling and micromachining of materials by laser pulses depends significantly on the aspect ratio (depth/width) of machined slots or cavities. If the aspect ratio increases the efficiency (ablation rate) drops due to re-deposition of ablated material on the cavity walls. At moderate laser intensities, significant fraction of the ablated material can be removed from the surface in the form of liquid droplets, e.g. [10, 14]. Removal of liquid droplets as well as their dynamics and re-deposition are of importance for prediction of both the ablation rate and the contamination of fabricated surfaces by debris of ablated material. Expansion of a laser plume induced by nanosecond laser pulses, however, was studied mostly for planar targets [4, 5, 7, 11, 13], and behavior of debris was not considered yet. Due to small diameter of laser spot and high rates of cooling, mixing, and expansion,

A.N. Volkov (B) Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904-4745, USA e-mail: [email protected]


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the flow in the plume is non-equilibrium. The Direct Simulation Monte Carlo (DSMC) is the most appropriate tool for simulations of gas flows in these conditions [7, 11, 13]. The purpose of this work is to develop a combined computational model for simulation of the two-phase laser plume expansion. The model is implemented in a parallel computational code and is applied for simulation of laser plume expansion from the bottom a cylindrical cavity in a silicon target into an ambient gas. The cavity is used as a model of a hollow formed by preceding laser pulses during multi-pulse micromachining of initially flat targets. This computational setup is intended for investigation of the efficiency of material removal depending on the cavity aspect ratio.

2 Combined Computational Model of Laser Ablation The combined computational model consists of three main parts, namely, a model of laser heating of the target material and its evaporation, a model of the expansion of the plume composed of Si atoms and molecules of an ambient gas, and the model of motion, heat and mass transfer of liquid droplets introduced into the flow from the surface of the molten target material. It is assumed that the heating of the material, plume expansion, and droplet motion are axisymmetric and considered in cylindrical coordinates Oxy (Fig. 1). The laser pulse is assumed to be Gaussian in time and space and intensity its I L (y, t) at time t is equal to I L (y, t) = Imax exp −t 2 / 2σt 2 − y 2 / 2σ D 2 , where

Imax = E L / (2π)3/2 σt σ D 2 , σt = t L /(8ln2)1/2 , σ D = D L /(8ln2)1/2 , E L is the pulse energy, t L and D L are the pulse duration and the laser spot diameter (FWHM), respectively. The model of the process is based on the assumption that the intensity of laser radiation I L is low enough so that (1) phase explosion does not take place, (2) ionization and dissociation effects and absorption of radiation by the plume are negligible and (3) the ablated material is an atomic vapor. Heating and evaporation of the target is described by the two-dimensional axisymmetric unsteady heat conduction equation and an additional equation with respect to the shape function of the target surface [13], which account for the latent heat of melting and for the changes in the shape of the target surface due to the material removal. The absorption of laser radiation by the target material is described by Beer’s law. The evaporation of the target material is calculated with the Hertz-Knudsen model [3]. In the kinetic model of the plume expansion, the components of the gas mixture (silicon vapor and surrounding gas) are assumed to be rarefied gases. Physical properties of a gas particle of sort A (A = Si for silicon atoms, A = G for ambient gas molecules) are described by its constant diameter dA , mass m A , and number ζA of internal degrees of freedom. The total cross-section for a collision between particles of sorts A and B is defined as σAB = π(dA + dB )2 /4, while changes in particle velocities and internal energies after a collision are described by the combined hard sphere – Larsen-Borgnakke model [1, 2]. The probability of energy

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exchange between translational and internal degrees of freedom is assumed to be equal to unity. The velocity distribution of evaporating atoms on the target surface is assumed to be half-Maxwellian in accord with the Hertz-Knudsen model, while all incident Si atoms are assumed to condensate on the surface. The interaction of the ambient gas molecules with the target surface is described by the model of diffuse scattering [3]. Submicron droplets are introduced into the pre-calculated, non-stationary flow field near the laser spot and then their Lagrangian trajectories are tracked according to equations of motion, heat and mass transfer dri = Vi , dt

dVi = f Di + fGi , dt dTi m i (c p + Hm δ(Ti − Tm )) = qi + Hb i , dt mi

(1) dm i = i , dt

(2)

where ri , Vi , Ti , and m i = (4/3)πρ p ri3 are the position vector, velocity vector, temperature, and mass of a spherical droplet of radius ri ; ρ p , c p , Tm , Hm , and Hb are the particle material density, specific heat, melting temperature, latent heat of melting, and latent heat of evaporation; fDi and fGi = m i gkx are the drag and gravity forces, qi , is the convective heat flux, and i is the vapor mass flux to the surface of the droplet; g is Earth’s gravity acceleration and kx is the unit vector directed along the axis Ox downwards in Fig. 1. The effect of droplets on the gas flow in the plume is not taken into account.

Fig. 1 Computational setup for simulation of the laser plume expansion from the bottom of a cylindrical cavity of diameter D and depth H on a silicon target into an ambient gas

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In the range of ambient gas pressure under consideration, the mean free path of gas molecules in an undisturbed ambient gas is about 0.1–1 μm, so that the effects of rarefaction of the flow around an individual submicron droplet on fDi , qi , and i need to be accounted for. The drag force is represented in its common form, e.g. [12], where the drag coefficient is calculated with equations given in [6]. The heat flux is represented in the form similar to that used in [12], where the Nusselt number is defined by an equation obtained in [8], but qi is assumed to be proportional to T − Ti0 , where T is the local gas temperature and Ti0 is the adiabatic temperature given by an equation in [9]. The mass flux is calculated with the approximate model i = 4πri2 [ψ(n, T )−ψ(n s (Ti ), Ti )], where ψ(n, T ) is the vapor mass flux density calculated with the half-Maxwellian distribution at number density n and temperature T, n is the local vapor number density, and n s (T ) is the number density of the saturated vapor at temperature T . In the beginning of every time step t k of the computational algorithm new droplets are introduced into the gas flow on the target surface. These droplets are assumed to be distributed along the target surface with an equal spacing y and have initial parameters ri = r0 , Vix = V0 , Viy = 0, Ti = Tw (yi , t k ), where Vix and Viy are x and y components of the droplet velocity vector, yi is the initial y coordinate of the droplet, Tw (y, t) is the surface temperature. The initial radius r0 and velocity V0 of droplets are considered as parameters of the model. The droplets are added into the flow only during the time, when the temperature of the target in the center of the laser spot is above the melting temperature Tm .

3 Computational Results In simulations, a flat silicon wafer with a cylindrical cavity of the aspect ratio H/D is considered (Fig. 1). The space above the target surface is filled by an ambient gas with initial pressure p0 and temperature T0 . The initial temperature of the target is equal to T0 . The bottom of the cavity is irradiated by a nanosecond laser pulse. The spot diameter D L is assumed to be much smaller than the cavity diameter D, so that the unsteady temperature field is calculated only for the material below the cavity bottom (at x > −HS in Fig. 1), while the temperature of the cavity wall is assumed to be constant and equal to T0 . Properties of solid and liquid silicon, and silicon vapor are given in [13]. In particular, Si optical properties are chosen for a third harmonic solid state laser with the wavelength 355 nm. Parameters of the laser pulse were equal to: t L = 23 nm, D L = 26 μm, and E L = 30.6 μJ. T0 was equal to 300 K, p0 was varied from 0.1 to 1 bar, the cavity diameter and depth were equal to D = 75 μm and H = 150 μm, respectively. Two types of surrounding gas were considered: Air (m G = 29 a.m.u., dG = 0.37 nm, ζG = 2) and freon HFC-134a (m G = 102 a.m.u., dG = 0.6 nm, ζG = 20). Initial radius of droplets r0 is varied from 0.03 to 0.3 μm, and the initial velocity V0 is assumed to be zero. The heat conduction equation for the target material is solved numerically by the implicit finite-volume scheme of second order in time and space. Calculations


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of the plume expansion were performed with the DSMC method based of the ‘No Time Counting’ scheme [1]. In order to decrease the level of statistical scattering, the computational results are averaged over ∼ 200 samples of the entire unsteady process. Equations (1)–(2) are integrated by the explicit Runge-Kutta method of the second order. In order to calculate instantaneous values of local gas parameters, a special algorithm for temporal and spatial interpolations is developed. The entire computational algorithm is implemented in a parallel computational code. The typical number of modeling gas particles was equal to 600 × 106 , typical number of cells of the computational mesh was equal to 30 × 106 . Simulations show that the mixing of the silicon vapor and the ambient gas is essentially non-equilibrium on the time scale of the plume expansion inside the cavity. The flow structure changes few times before the shock wave leaves the cavity. At the initial stage of the plume expansion, the maximum gas mixture temperature is realized near the cavity bottom, however, by 30 ns the maximum temperature is observed near the cavity wall, where a few shock waves interact with each other (Fig. 2a). The radial size of this high-temperature region increases in time, while the intensities of secondary shock waves drop, and by the time 70 ns (Fig. 2b) the maximum temperature is observed on the flow axis, while all secondary shock wave disappear. Later on, This high temperature region flows up and is cooled down (Fig. 2c). Analysis of the transient vapor field shows that large number densities of the vapor are realized only inside the cavity and in the hemisphere of diameter ∼1.5D around the cavity. The vapor cloud is found to be effectively frozen in shape and size by the time when the shock wave reaches the distance about 0.75D from the target

Fig. 2 Instantaneous mixture temperature fields during the laser plume expansion in the cylindrical cavity with aspect ratio H/D = 2 into air with pressure 0.1 bar. Panel (a) corresponds to time 30 ns; (b), 70 ns; (c), 140 ns. The time scale is relative to the peak of the Gaussian-shaped temporal laser pulse

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surface. This behavior of the vapor cloud is similar to that realized in the plume expanding from a planar target [13]. It is observed only if the ambient gas pressure is large enough and is caused by the formation of a stagnation region inside and near the cavity, where both axial and radial components of the mixture velocity are fairly small. Later on, the vapor density in the cloud slowly drops due to molecular diffusion of Si vapor into the ambient gas, vapor condensation on the target surface, and formation of Si clusters.

Fig. 3 Instantaneous patterns of inertial less markers (a, d) and droplets of initial radii 0.1 μm (b, e) and 0.3 μm (c, f) during plume expansion in the cavity of aspect ratio H/D = 2. Panels (a, b, c) correspond to expansion into air at time 150 ns. Panels (d, e, f) correspond to expansion into freon at time 350 ns. Ambient gas pressure is equal to 0.1 bar


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The plume expansion into freon HFC-134a is characterized by the slower motion of the shock wave, which velocity is roughly twice smaller than that in air because of smaller sound speed in freon as compared with air. It was found that only droplets with initial radius smaller that than 0.1–0.2 μm can leave the cavity at least in the case when droplets are generated on the cavity bottom with zero initial velocity (Fig. 3). The conditions for droplet removal from the cavity are somewhat more favorable in freon than that in air because the time interval, when the gas flow accelerates droplets, is larger in freon. In freon surroundings, the density of the gas mixture is also larger than that in air, and the particle relaxation time (Stokes number) becomes smaller and this also favors the acceleration of the droplets by the plume. To a rough approximation, the particles are accelerated by the flow until the time, when the shock wave leaves the cavity. Following this, the droplets decelerate in the stagnation region formed inside and near the cavity. The radial motion of droplets is relatively weak, which suggests that the re-deposition of droplets far from the cavity can be induced by the deviation of the real flow from the idealized axisymmetric expansion, and, perhaps, initial radial velocities acquired by droplets during the fragmentation of the target material.

4 Conclusion A combined computational model for simulation of two-phase plume expansion from a bottom of a cylindrical cavity induced by a nanosecond laser pulse into an ambient gas is developed and implemented in a parallel computer code. The model can be used for prediction of the material removal during laser micromachining in the form of both atomic vapor and submicron debris. Acknowledgements The financial support is provided by the EU Marie Curie programme, project MTKD -CT-2004-509825.

References 1. Bird, G.A.: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford (1994) 2. Borgnakke, C., Larsen, P.S.: Statistical collision model for Monte Carlo simulation of polyatomic gas mixture. J. Comp. Phys. 18, 405–420 (1975) 3. Cercignany, C.: Rarefied Gas Dynamics: From Basic Concepts to Actual Applications. Cambridge University Press, Cambridge (2000) 4. Chen, Z., Bogaerts, A.: Laser ablation of Cu and plume expansion into 1 atm ambient gas. J. Appl. Phys. 97, 063305 (2005) 5. Gusarov, A.V., Smurov, I.: Near-surface laser – vapour coupling in nanosecond pulsed laser ablation. J. Phys. D 36, 2962–2971 (2003) 6. Henderson, C.B.: Drag coefficients of spheres in continuum and rarefied flows. AIAA J. 14, 707–708 (1976) 7. Itina, T.E., Hermann, J., Delaporte, P., Sentis, M.: Laser-generated plasma plume expansion: Combined continuous-microscopic modeling. Phys. Rev. E. 66, 066406 (2002)

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8. Kavanau, L.L.: Heat transfer from spheres to a rarefied gas in subsonic flows. Trans. ASME. 77, 617–623 (1955) 9. Koshmarov, Y.A., Svirshevskii, S.B.: Heat transfer from a sphere in the intermediate dynamics region of a rarefied gas. Fluid Dyn. 7, 343–346 (1972) 10. Porneala, C., Willis, D.A.: Observation of nanosecond laser-induced phase explosion in aluminum. Appl. Phys. Lett. 89, 211121 (2006) 11. Sibold, D., Urbassek, H.M.: Effect of gas-phase collisions in pulsed-laser desorption: A threedimensional Monte Carlo simulation study. J. Appl. Phys. 73, 8544–8551 (1993) 12. Volkov, A.N., Tsirkunov, Yu.M., Oesterle, B.: Numerical simulation of a supersonic gas-solid flow over a blunt body: The role of inter-particle collisions and two-way coupling effects. Int. J. Multiphase Flow. 31, 1244–1275 (2005) 13. Volkov A.N., O’Connor, G.M., Glynn, T.G., Lukyanov, G.A.: Expansion of a laser plume from a silicon wafer in a wide range of ambient gas pressures, Appl. Phys. A 92, 927–932 (2008) 14. Yoo, J.H., Jeong, S.H., Greif, R., Russo, R.E.: Explosive change in crater properties during high power nanosecond laser ablation of silicon. J. Appl. Phys. 88, 1638–1649 (2000)

Study of Dispersed Phase Model for the Simulation of a Bubble Column F. Sporleder, Carlos A. Dorao, and H.A. Jakobsen

Abstract The variations in the size and shape distributions of the dispersed phase play a major role in the simulation of bubble column flows. The population balance equation (PBE) can be used to describe the evolution of the dispersed phase. Nevertheless, this is computationally demanding. The present work extends previous work by applying a spectral element method of a least squares type to solve this equation when studying three-dimensional transient multi-phase flow. The analysis is focused on the use of a velocity distribution which depends on the bubble size.

1 Introduction It is of interest to account for the variations in the size and shape distributions of the dispersed phase when simulating bubble column reactor flows. Typical twofluid models are not able to handle this complex phenomena because only one characteristic size for the dispersed phase is used. The PBE [7] uses probability density functions to account for variations in the properties of the bubbles. It is a non-linear partial integro – differential equation, and is computationally costly [6]. Recently, Dorao and Jakobsen [1] showed the applicability of the Least-Squares Method (LSM) [4] to solve the PBE. Patruno et al. [5] presented a novel multifluid framework using population balance, where the dispersed phase velocity is a function of the internal coordinate. The present work is part of the development of a numerical tool based on the least-squares spectral element method to simulate transient bubbly flow in a 3D cylindrical domain, based on the multi-fluid framework presented in [5]. The focus of this analysis is set on the concept of using a velocity distribution dependent on the particle size. Furthermore, the concept of a growth velocity is introduced. This velocity appears in the general PBE, and accounts for changes in the size of the particles due to phenomena such as pressure variations.

F. Sporleder (B) Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway e-mail: [email protected]


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2 Mathematical Model and Numerical Method Equation (1) is the general form of the PBE [6] ∂vξ (x, ξ, t) f ∂f + ∇x · (u(x, ξ, t) f ) + = (L B + L D ) f ∂t ∂ξ

(1)

where f is the distribution function, u = (vx , v y , vz ) is the velocity

number density ∂ , and vξ is the growth velocity.The operators L D and L B field, ∇x = ∂∂x , ∂∂y , ∂z represent the death and birth rates of the particles given by phenomena such as coalescence and breakage. Some of these are integral terms. Equation (1) is solved using the Least Squares Spectral Element Method (LSSEM) [4]. This method presents attractive characteristics, such as exponential convergence rates and a posteriori error estimators. It is independent of the equations at hand, and is capable of solving integro-differential equations. Furthermore, a wellposed formulation always yields SPD matrices [2]. In the present study, we used Lagrange polynomials with nodal points at the zeros of the Gauss-Lobatto-Legendre (GLL) integration rule. We used a Crank-Nicholson discretization scheme to approximate the time derivative [4]. We also took the internal property ξ to be the diameter of a bubble, such that its volume is defined as π6 ξ 3 . The void fraction can be calculated as the integral of f in ξ .

3 Numerical Experiments Some numerical experiments are presented in this section. For simplicity, we neglect the birth and death source terms. All experiments are performed using a cylindrical grid such as the one shown in Fig. 1. The cylindrical cross-section was generated following [3].

(a)

(b)

Fig. 1 Example of the grid and its x − y cross section used in the numerical experiments

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3.1 Test of Numerical Scheme We considered constant axial and growth velocities, and null velocities in the other directions. The initial condition was set as follows: ⎞ ⎛ (ξ −0.03)2 (z−0.2)2 − − 1 2 2σξ2 (2) − β⎠ f (t=0) = √ e 2σz − α ⎝e 2π σξ

10−1

t=1 t = 2.5 order 2

10−2 10−3 10−4

10−2 10−1 Time interval [s] (a)

100

t = 1s Ne = 5 t = 1s Np = 5 t = 2.5s Ne = 5 t = 2.5s Np = 5

−1

10

10−2 101 Number of Interpolants/Elements (b)

L2 norm of residual

100

L2 norm of difference

L2 norm of difference

with σz = .05m, σξ = .007m, vz = 0.2m/s and vξ = 0.01m/s . The coefficients α and β are defined such that f d(z=0) = f d(ξ =0) = 0. Convergence is measured with the L 2 norm of the difference between the approximate solution and the exact solution. Additionally, the norm of the residual of the equation is analyzed when varying the approximation parameters of the LS-SEM. Figure 2a evidences the second order of the Crank-Nicholson scheme used for the time derivative. Figure 2b, c show the exponential convergence rate obtained with the LS-SEM. Nevertheless, the maximum accuracy that can be obtained is dictated by the time discretization.

10−2

t = 2.5s Ne = 5 t = 2.5s Np = 5

10−3 10−4 101 Number of Interpolants/Elements (c)

Fig. 2 Variation of the L 2 norm of (a)−(b) difference and (c) residual with the size of the time step in (a) and with the number of interpolants and elements in (b)−(c).

3.2 Trajectory of a Bubble The first application we present is a single rising bubble. We used the measured instantaneous velocities in [9] as input to our model. Growth velocity was neglected. Equation (3) shows the initial condition used, with σ = 0.004m.

f (t=0) = e

−

x 2 +y 2 +(z−μz )2 2σ 2

(3)

We considered f (x=0) = 0. Figure 3 compares the results obtained with the measured coordinates in [9]. The simulation shows good agreement with the experimental results.

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0.065

5

0.06

4

0.055 Axial position [m]

Position [m]

6

3 2 1 0

x x − Tomiyama (2002) y y − Tomiyama (2002)

−1 −2

0

0.05

0.1 Time [s]

0.15

z z − Tomiyama (2002)

0.05 0.045 0.04 0.035 0.03 0.025 0.02

0

0.05

0.1 Time [s]

0.15

0.2

(b)

(a)

Fig. 3 Coordinates of the bubble as a function of time

3.3 Bubble Plumes The second application analyzes how diameter-depending velocities can influence the results obtained. Here, we took the experiments in [8], which analyze three plumes of different bubble sizes and how they grow with time. We only considered the terminal velocity v∞ mentioned in [8], assuming stagnant liquid and no walls-effects for an air-water system at STP. The initial and boundary conditions were set as follows: 2

−y2 f (z=0) = h (−0.1,μ1 ,σ1 ) + h (0,μ2 ,σ2 ) + h (0.1,μ3 ,σ3 ) e 2σx −z 2 2

f (t=0) = f (z=0) e 2σx with:

− (x−μ2x ) 1 h (μx ,μ,σ ) = √ e 2σx 2π σ 2

(4) 2

− (ξ −μ) 2

2

2σ

where σx = .01, μ1 = .0015, σ1 = .0004, μ2 = .0005, σ2 = .00015, μ3 = .003, and σ3 = .0005, in SI units. As shown in Fig. 4a, the plumes reach the top at different times. It is worth noticing that after 3 s a plume of bubbles of 0.5 mm of diameter should have traveled a distance of approximately 0.16 m. Nevertheless, both the experiments and our calculations show that most of the gas remains at the first third of the column (0.6 m) at that time (see Fig. 4c). This has to do mainly with the fact that the bubbles injected have a range of sizes, and while the bigger ones ascend faster, the smaller ones move more slowly and gather with the newer bubbles. This difference is well reflected in the void fraction for our calculations, and are in good agreement with the experiments.


(a) t = 2s

(b) t = 3s

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(c) t = 3s

Fig. 4 Void fraction contours after 2 and 3 s for y = 0. Figure (c) was extracted from [8], and reprinted with permission of ASME

3.4 Growth Velocity The final numerical experiment was focused on the influence of the growth velocity on the void fraction. Assuming that a bubble conserves its mass and its density ρ is time – independent, then the growth velocity is given by vξ = −ξ 3ρ u · ∇x ρ. Here, the terminal velocity mentioned in Sect. 3.3 was used, and the other velocities were considered to be zero. The initial condition used is shown in Eq. (5).

f (t=0)

2 2 16 +y (ξ −.02)2 1 − x .25 2 − σ − (z−.4) ξ e .02 − α = −β e √ 2π σξ

(5)

where σξ = 0.006. The constants α and β were set so as to have f (z=0) = f (ξ =0) = 0. For this case, a 4m height was considered for the tube. Figure 5 shows the results obtained for cases with and without growth velocity. Bubbles move with different velocities according to their size, following the terminal velocity imposed. The growth velocity affects more strongly the larger bubbles than the smaller ones. This causes an extension of the region where bubbles are found, and reduces the maximum value of the void fraction.

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0.27 0.25 0.2 0.15 0.1 0.05 0.01 0.1 0.25 0.5

0.4

0.01

0.01 0.9 0.45 0.1 WITHOUT (a)

WITH

WITHOUT (b)

WITH

WITHOUT

WITH

(c)

Fig. 5 Comparison of the void fraction with and without growth velocity. It shows the contour plots for the void fraction at different times, for y = 0. (a) t = 0 s, (b) t = 4 s, (c) t = 8.5 s

4 Summary In this work, the PBE in 4D plus time was presented. The resulting model was solved using the LS-SEM, with a Crank-Nicholson scheme for time. Numerical examples were presented for showing the capabilities of the numerical tool developed. In particular, we showed that treating the size of the bubbles as a continuum, the void fraction is affected considerably. Furthermore, the growth velocity term that appears on the presented model influences appreciably the void fraction for large vertical problems. Acknowledgements The PhD fellowship (F. Sporleder) financed by the Research Council of Norway through the GASSMAKS Program is gratefully appreciated.


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References 1. Dorao, C.A., Jakobsen, H.A.: hp-adaptive least squares spectral element method for population balance equations. (2008). Appl Num Math, doi: 10.1016/j.apnum.2006.12.005 2. Gerritsma, M., De Maerschalck, B.: Least-squares spectral element methods in computational fluid dynamics. In: Koren B., Vuik C.(eds.) Advanced Computational Methods in Science and Engineering, Lecture Notes in Computational Science and Engineering 71. Springer, Berlin (2009). doi:10.1007/978-3-642-03344-5_7 3. Gordon, W. J., Hall, C. A.: Construction of curvilinear co-ordinate systems and applications to mesh generation. Int. J. Numer. Methods Eng. 7, 461–477 (1973) 4. Jiang, B.-N.: The least-squares finite element method: theory and applications in computational fluid dynamics and electromagnetics. Springer Heidelberg (1998) 5. Patruno, L.E., Dorao, C.A., Svendsen, H.F. Jakobsen, H.A.: On the modeling of droplet-film interaction considering entrainment, deposition and breakage processes. Chem. Eng. Sci. 64(6), 1362–1371 (2009) 6. Patruno, L.E.: Experimental and numerical investigations of liquid fragmentation and froplet generation for gas processing at high pressures. Ph.D. thesis 2010:111. Norwegian University of Science and Technology, Norwegian (2010) 7. Ramkrishna Population Balances, Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego, CA (2000) 8. Tomiyama, A., Shimada, N.: A Numerical Method for Bubbly Flow Simulation Based on a Multi-Fluid Model. J. Pressure Vessel Technol. 123, 510–516 (2001) 9. Tomiyama, A., Celata, G., Hosokawa, S., Yoshida S.: Terminal velocity of single bubbles in surface tension force dominant regime. Int. J. Multiphase Flow 28, 1497–1519 (2002)

Application of Invariant Turbulence Modeling of the Density Gradient Correlation in the Phase Change Model for Steam Generators Yasuo Obikane and Shigeru Ikeo

Abstract Predictions of the amount of steam generated on evaporation plates are mostly done by empirical algebraic models or semi-algebraic one equation models. However, the accuracy of the methods fully depends on the equipment and the environment. A new method that directly installs the phase change model into the system of fluid dynamics equations is studied for improvement. First, a turbulence model for the phase field model below the boiling point or near the boiling point is constructed. The model consists of the density gradient correlations, which are shown in the set of the phase field equation (C-H equation). The density gradient correlation is obtained with the Reynolds decomposition from the compressible continuity equation. Second, to import the effect of viscosity, the pseudo stress tensor in the momentum equation is modified in a semi-direct simulation, which covers mainly large eddies on coarse grids. A steam generator is modeled with a three-layer problem: the water layer, the humid air layer, and the environmental boundary in nearly homogeneous turbulence with strong acoustical disturbances, which are common in the multi-phase flows in pipes. The density gradient correlation equation is closed by the invariant modeling technique with many unknown constants. In the present work, one constant appearing in the invariant model is determined theoretically. The set of equations allows stable computations during prediction for the steam generator.

1 Introduction In predicting the efficiency of a steam generator, the information on the generation of steam in turbulent flows is important. This research aims to describe a complex multi-phase flow with the phase field model (PFM) in turbulent flows, which will help to predict the evaporation process in more detail. This method is superior at predicting steam generation than the classical semi-experimental method, which Y. Obikane (B) Institute of Computational Fluid Dynamics, Meguroku-ku, Tokyo 152-0011, Japan; Department of Mechanical Engineering, Sophia University, Tokyo 102-8554, Japan, e-mail: [email protected]; [email protected]


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requires many experimental constants to fit the computations. However, in the new method tedious checking is necessary in order to conduct the steps for computations. First we define the mass flux through the interface by the integral formation. To describe the complicated effect of the phase change and the variation of density, we introduce the turbulence pressure tensor with both normal and cross correlations of the density fluctuating terms in the C-H equation which can be modeled. For the present study the same invariant closure technique by Y. Obikane in [1] is used. The anisotropy tensor of the movement of the fluctuating part of the density gradient is used to model the equation. As the temperature of the steam generator is under the boiling point, it is assumed that the diffusion function in the PFM only depends on the order parameter. So, the present PFM is simpler than the usual models [2]. The computation is done with TVD in the combined differential system of the compressible flow, the PFM equation, and the invariant turbulence model.

2 The Governing Equation of the System of the Phase Field Model for Compressible Flows The differential system of the compressible Navier Stokes equations can be written as ⎛

−ρu i ⎜ −ρu u + P − τi j i j i j ⎜ ∂ −εu i + (−Pi, j + τi, j ),j u i + κ T, j − ks ρu i,i ρ,u j ), j X =⎜ ⎜

∂t ⎝ ∂ −u i φ,i + Γ0 φ ∂φ Ψ − k2 φ,qq ,i

,i

⎞

⎛ ⎟ ⎟ ⎜ ⎟ +⎜ ⎟ ⎝ ⎠

⎞ − dm dt 0⎟ ⎟ dm C Latent ⎠

(1)

dt

0

,i

where i=1,2,3 and j=1,2,3, X=(ρ, ρu i , ε, φ), and where Γ0 is a positive constant and represents the mobility intensity. The pressure tensor Pi j is defined as 1 Pi j = (P − κ(ρρ,qq + (abs(ρ,q )2 )δi, j + κs ρ,i ρ, j 2

(2)

where κ is the surface tension parameter. If there is a density gradient, then

a surface tension of σ with κs is computed. The total energy ε is defined as ρ e + 12 u i u i , and the state equation is defined as p = ρ RT , and ρ(φ) = F(φ).

3 Evaluation of Evaporation Rate Since the present computation is conducted below the boiling point, the model has been modified. Provided that there is no mass flux inside the domain, the evaporation rate dm/dt through the interface is modeled by the following equation:

Application of Invariant Turbulence Modeling

dm =C dt

∂ρ ∂φ

123

(φ Saturated − φ Local )

(3)

where the order parameter φ Local is taken as less than the saturated order parameter. The coefficient C is a function of both convection speed and temperature, and can be expressed as C = Cconvection + Cther mal . Furthermore, from Ref [3], if the temperature increases, then the width of the interface between vapor and water increases, and the gradient of the order parameter in the interface decreases. Thus, for the computation of the steam generator, the thickness of the interface in the boundary layers on the evaporation plates is used to indicate the rate of evaporation, i.e., the rate of the vaporization from the flat plates is assumed to be proportional to the rate of change of the thickness of the order parameter.

4 A Turbulence Model of the Fluctuating Density Gradient (FDGC) for the Phase Field Modeling Denote the mean density and the fluctuating density as ρˆ and ρ. ´ Then, the mean value of the pressure tensor Pi j is decomposed into the mean value and fluctuating parts as: 1 Pî j = ( p − κ(ρρ,qq + (abs(ρ,q )2 )δi, j + κs ρ,i ρ, j + δ Pi j 2

(4)

where p is the static pressure and δ Pi j is the moment of the fluctuating parts defined as 1 δ Pi j = κ ρ´,q ρ´,q − (ρρ,q ),q − abs(ρ,q )2 δi, j + κs ρ´,i ρ´, j (5) 2 It should be mentioned that most terms contain only the density gradient correlation.

5 PFM Model in Turbulence The formal definition φ of the PFM model is given as: ∂Ψ ∂φ = Γ0 φ − k2 φ,qq ∂t ∂φ ,i

(6)

,i

where the chemical potential depends on the order parameter and the local temperature T. The following derivative is interpreted as a diffusion function ζ : ζ (φ, T ) =

∂Ψ = F(φ, T ) ∂φ

(7)

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´ Equation (1) then becomes: The PFM equation can be decomposed as φ = φˆ + φ. ∂ φˆ ∂ ˆ = −uˆ i φˆ ,i + Γ0 φˆ Ψ − k2 φˆ ,qq + δF R ∂t ∂ φˆ ,i

(8)

,i

The fluctuating part δ F R can be expanded as: 1 ´ ,ii j j − uí φ´,i δ F R = Γ0 (φ´ ,i ζ´ ) − Γ0 K 2 ((φ´ ,i φ´ ,i ), j j + φíi φˆ, j j − (φ´ φ) 2

(9)

Since turbulence is diffusive, δ F R must be negative all the time. Thus, the model must satisfy the following inequality: δF R < 0

(10)

The key to the work is as follows: The φ´,i φ´, j can be replaced by the one-to-one relation between φ and the density ρ. Thus, Eq. (6) can be rewritten with the following relation: φ´,i φ´, j =

∂φ ∂ρ

2 ρ´,i ρ´, j = c2 ρ´,i ρ´, j = c2 Di j

(11)

where the fluctuating density gradient correlation (FDGC) is denoted as Di j . In this way, the present problem is deduced to solve FDGC. It should be mentioned that the total density is defined as: ρ = ρair + ρv , ρv = ρw f (φ), and ρv < ρ Sat (T ) where ρw is the density of water. Since the total density is calculated by the continuity equation and the vapor density is determined by the PFM equation, the density of air is determined by the above equations. The correlation of the fluctuating density used in the phase field model is also obtained by the Reynolds decomposition as: ρ´,i ρ´, j = ρair,i ´ ρair, ´ j + ρv,i ´ ρv, ´ j + ρv,i ´ ρair, ´ j

(12)

Since non-vapor perturbation means non-density fluctuation in the present problem, ´ ρv, ´ j . To model the equation, it is convenient to it can be assumed that ρ´,i ρ´, j = cρv,i use the anisotropy tensor for the fluctuating density gradient: di j = (ρ,i ρ, j )/(ρ,i ρ,i ) − 1/3δi j

(13)

and the incompressible anisotropy tensor as bi j = (u i u j )/(u i u i ) − 1/3δi j This can be reformed to the frozen turbulence equation in the case of strong shear and deformation as follows:


∂ ρ,q ρ,k = −(ρ,q u i,ik + ρ,k u i,iq )ρˆ − ρ,q ρ,i uˆ i,k − ρ,k ρ,i uˆ i,q ∂τ = −(αd (Id , I Id , I I Id )dqk − αb (Ib , I Ib , I I Ib )bqk )ρˆ −ρ,q ρ,i uˆ i,k − ρ,k ρ,i uˆ i,q

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(14)

The invariant terms αd vanish if there is no density variation.

6 Setting Numerical Experiments The model of the steam generator has one separator in a channel, so it has four surfaces with sources of water. The boundary values of φ on the wall are set to 0.8 and the value of φ at the inlet varies from 0.1 to 0.25 for the parameter study. The inlet speed of humid air is 3 m s−1 , while the length of the flat plate is 0.14 m. The physical parameters are ck2=0.02 and γ = 0.1, which were obtained in the

9.00E-01

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Fig. 1 Change of thickness of the interface

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disturbed flow analysis by Y. Obikane in [3]. The changes of the thickness of the order parameter are examined on the flat plates.

7 Results of Numerical Experiments In Figs. 1, 2, the thickness of the interface of the order parameters is about 9 mm at the outlet, and the slopes are getting smaller with larger values of φ at the inlet. This shows that the interface is diffused by convection. The relation between the thickness of the order parameter and density could be used to evaluate the real evaporation rate using the Eq. (3). Since the temperature is fixed in the model, if ∂ρ the initial value of the saturated vapor and the evaporation rate are set, then ∂φ , and the coefficient C conv is obtained with the computed thickness, which includes the effect of the convection speed. The computation does not diverge in the present case if CFL is taken at order 10−3 .

8 Conclusion The present method can compute the development of the interface break-up along the boundary layer in perturbations by two-dimensional semi-DNS with the turbulence model. However, more research is necessary to control for all phenomena within this model to apply to the design of real steam generators.

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Acknowledgements The Authors would like to thank Dr.N. Takada, AIST (National Institute of Advanced Industrial Science and Technology) for his valuable comments.

References 1. Takada, N., Tomiyama, A., Hosakawa, S.: Numerical simulation of drops in a shear flow by a lattice-Bolzmann binary fluid model, Comput. Fluid Dyna. J. 12,475–481 (2003) 2. Obikane, Y.: Research on the correlation of the fluctuating density correlation of the compressible flows. WASET Proc. 60(137), 79–793 (2009) 3. Obikane, Y.: A frequency dependence of the phase field model in Laminar Boundary Layer with periodic perturbations. WASET Conference Paper,ICCTGF10, Tokyo, JP65000 (May, 2010)

Part V

Algorithms

Reformulated Osher-Type Riemann Solver Eleuterio F. Toro and Michael Dumbser

Abstract We reformulate the Osher Riemann solver by, first, adopting the canonical path in phase space, and then performing numerical integration of a matrix. We compare the reformulated scheme of this chapter with the original Osher scheme on a series of test problems for the one-dimensional Euler equations for ideal gases, concluding that the present solver is simpler, more robust, more accurate and can be applied to any hyperbolic system.

1 Introduction Finite volume and discontinuous Galerkin methods for hyperbolic conservation laws require a numerical flux, for which there are essentially two approaches to obtain it, namely the centred and the upwind approaches. Centred fluxes do not explicitly use wave propagation information while upwind schemes explicitly use wave propagation information via the solution of the local Riemann problem, solved exactly or approximately. Centred schemes are simpler and more general than upwind schemes. However upwind schemes are more accurate, provided the Riemann solver used is complete, that is the structure of the solution, or wave model, contains all the characteristic fields present in the exact problem. Riemann solvers that do not have this property, called incomplete, have a similar performance to centred methods, characterised by excessive numerical dissipation for waves associated with intermediate charateristic fields. Usually these are much harder to compute accurately than the faster waves, such as shock waves. Intermediate charateristic fields are associated with important physical features such as contact discontinuities, ignition fronts, material interfaces and vortices. Complete Riemann solvers include the exact solver [5], naturally, and the Osher [4] solver. E.F. Toro (B) Laboratory of Applied Mathematics, Department of Civil and Environmental Engineering, University of Trento, I-38100 Trento, Italy e-mail: [email protected]


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This paper adopts the upwind approach based on a complete Riemann solver so as to better resolve waves associated with intermediate characteristic fields. To this end we build upon the Osher Riemann solver, which is non-linear and complete. Additional attractive features of the Osher approach are robustness, entropy satisfaction, good behaviour for slowly-moving shocks and smoothness, property that makes it very attractive to the aeronautical community. However, the Osher solver is indeed very complex and probably for this reason it has remained unattractive to the scientific community interested in solving very complex hyperbolic systems. In this chapter we present a new version of the Osher scheme that is much simpler than the original one, making it applicable in fact to any hyperbolic system. If the complete eigenstructure of the system is available analytically, then the implementation of the method is straightforward. Otherwise one must first compute the eigenstructure numerically using standard software and then use it in the present scheme to determine the numerical flux. Here we illustrate the performance of the new scheme, in first-order mode, for the Euler equations of gas dynamics for ideal gases. In [1] and [2] the scheme, in first and high order mode, is applied to very complex hyperbolic systems. The rest of this chapter proceeds as follows. In Sect. 2 we present the reformulated Osher scheme. In Sect. 3 we show numerical results and conclusions are drawn in Sect. 4.

2 The New Osher-Type Solver We first give the necessary background and notation, and then present the details of our scheme. Consider a general m × m hyperbolic system of conservation laws ∂t Q + ∂x F(Q) = 0 ,

(1)

with the vectors of conserved variables and fluxes respectively denoted as Q = [q1 , q2 , . . . , qm ]T ,

F = [ f 1 , f 2 , . . . , f m ]T .

(2)

The real eigenvalues, written in increasing order, are denoted by λi (Q) and the corresponding right eigenvectors by Ri (Q), for i = 1, 2, . . . , m. Here we consider Godunov-type schemes of the form Qin+1 = Qin −

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(3)

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⎧ ⎨ Q0 if x < 0 , ⎩

(4) Q1 if x > 0 .

At present there are several ways of solving (1), (4) and computing Fi+ 1 . For back2 ground see [5] and references therein. Now recall that hyperbolicity of system (1) is equivalent to saying that the Jacobian matrix A(Q) of the flux F(Q) is diagonalizable, that is A(Q) = R(Q)Λ(Q)R−1 (Q) ,

(5)

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(6)

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(7)

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(8)

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(9)

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which will be directly used to calculate the Osher-type numerical flux of this chapter, as seen below. The Osher numerical flux [4] may be defined as Fi+ 1 = 2

1 1 (F(Q0 ) + F(Q1 )) − 2 2

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(10)

Q0

This requires the evaluation of an integral in phase space, which depends on the integration path chosen joining Q0 to Q1 . Originally, Osher proposed two ways of choosing integration paths so as to make the actual integration tractable, called the P-ordering and the O-ordering. However, the analytical calculations are still too involved to be performed for general hyperbolic systems. Full details of the original Osher Riemann solver are found in Chap. 12 of [5]. Here we first propose to select the canonical path ψ(s; Q0 , Q1 ) = Q0 + s(Q1 − Q0 ) ,

s ∈ [0, 1]

(11)

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to evaluate the integral in (10). Then, under a change of variables we obtain ⎛ Fi+ 1 = 2

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0

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Note that A(ψ(s j ; Q0 , Q1 )) must be decomposed as in (9) for each s j .

3 Numerical Results and Discussion Here we compare the original (analytical) Osher scheme with the reformulated (numerical) version of this chapter for problems with exact solution. To this end we solve the compressible Euler equations for an ideal gas with γ = 1.4. Table 1 gives the initial conditions for six test problems (Riemann problems) taken from [5]. Results are from the first-order version of the scheme (3), so that the assessment is focused only on the performance of the Riemann solver. High-order extensions following the ADER approach are presented in [1, 2]. Figure 1 shows the results for density for all six tests. From top to bottom, the left column shows results for tests 1, 3 and 5, while the right column shows results for tests 2, 4 and 6. For Test 1, which contains a sonic point, the classical Osher scheme with both P-ordering (not shown) and O-ordering gives equivalent results and these are equivalent to those of the present reformulated Osher scheme. For Test 2 the classical Osher scheme with P-ordering gives equivalent results to the present scheme but fails with O-ordering. For Test 3 both orderings in the Osher scheme give equivalent results (P-ordering not shown), also equivalent to those of Table 1 Initial left (L) and right (R) states for the density ρ, velocity u and the pressure p. Here tend is the output time and x d is the position of the initial discontinuity in the domain [0, 1] Case ρL uL pL ρR uR pR tend xd RP1 RP2 RP3 RP4 RP5 RP6

1.0 0.0 1.0 0.75 1.0 0.0 5.99924 19.5975 1.0 −19.59745 1.0 2.0

1.0 1.0 1, 000 460.894 1000.0 0.1

0.1 0.0 0.125 0.0 1.0 0.0 5.99242 −6.19633 1.0 −19.59745 1.0 −2.0

1.0 0.1 0.01 46.095 0.01 0.1

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Fig. 1 Results for the six test problems of Table 1 for the ideal Euler equations with γ = 1.4. Comparison between the present scheme with the original Osher solver and the exact solution. Results are shown for density for the first-order version of the schemes. In all computations the CFL coefficient was set to 0.9, with the domain of unit length discretized with 100 cells

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the present scheme. The same is true for Test 4, which contains a very slowly moving shock (left discontinuity). For Test 5 the Osher scheme with P-ordering fails miserably, while the O-ordering works and gives almost identical results to those of the present scheme, the latter being slightly better. For Test 6 the Osher scheme with O-ordering fails, while with the P-ordering works and gives similar results to those of the present scheme, noting however that the present scheme gives virtually non-oscillatory solutions behind the two shocks. In conclusion, the classical Osher scheme depends very crucially on the choice of path ordering and there is no single choice that works well for all problems considered here. Moreover, the failing of the classsical Osher scheme is catastrophic, see results for Tests 2, 5 and 6. The present version of the Osher scheme has retained all the good properties of the classical Osher scheme when this works. In this case our results are equivalent or slightly better to those of the classical Osher scheme. The present scheme works very well for all cases considered. It is also worth saying that the present scheme recognizes exactly a stationary contact discontinuity for the Euler equations. This is an important property of a numerical scheme not satisfied by incomplete Riemann solvers, such as HLL [3], or centred schemes, such as FORCE [6, 7].

4 Concluding Remarks We have presented a reformulated version of the Osher Riemann solver, which is simpler, more accurate and more robust than the original scheme. Due to its simplicity the new scheme is applicable to any hyperbolic system. Acknowledgements The research presented here was partially funded by the Italian Ministry of University and Research (MIUR) in the frame of the project PRIN 2007.

References 1. Dumbser, M., Toro, E.F.: A simple extension of the Osher Riemann solver to non–conservative hyperbolic systems. J. Sci. Comput. (2010). DOI: 10.1007/s10915-010-9400-3 (in press) 2. Dumbser, M., Toro, E.F.: On universal Osher–type schemes for general nonlinear hyperbolic conservation laws. Comm. Comput. Phys. (2011) (accepted for publication) 3. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov–type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983) 4. Osher, S., Solomon, F.: Upwind difference schemes for hyperbolic conservation laws. Math. Comp. 38(158), 339–374 (1982) 5. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition. SpringerVerlag, Berlin Heidelberg (2009) 6. Toro, E.F., Billett, S.J.: Centred TVD Schemes for hyperbolic conservation laws. IMA J. Num. Anal. 20, 47–79 (2000) 7. Toro, E.F., Hidalgo, A., Dumbser, M.: FORCE schemes on unstructured meshes I: Conservative hyperbolic systems. J. Comput. Phys. 228, 3368–3389 (2009)

Entropy Traces in Lagrangian and Eulerian Calculations Philip L. Roe and Daniel W. Zaide

Abstract We investigate the generation and propagation of errors in shockcapturing codes. We demonstrate that one type of error derives from the nonlinearity of the Hugoniot curve, but that this is not sufficient to explain all errors.

1 Introduction The decision to capture rather than fit shockwaves and other discontinuities probably compromises some aspects of a computation, although exactly how has never been completely clarified. One issue that has long been identified is that of the propagating errors called entropy traces by Rodzhestvenskii and Yanenko [6]. These are regions of spurious entropy or temperature that propagate along particle paths or solid boundaries. They are sometimes associated especially with Lagrangian methods, although we have found that nearly identical behavior can occur in Eulerian codes. For example, Fig. 1 compares Eulerian and Lagrangian computations of the Sod test problem. The Eulerian computation is done in a moving frame that makes the contact discontinuity stationary. Based on this and other experiments we have concluded that the mechanisms that create such traces are essentially the same for both types of code. Merely, Lagrangian methods suf fer the faults of their virtues. By using a mesh that follows the flow, diffusion across material pathlines can be reduced or eliminated. Then the good news is that physical changes to the entropy are conserved; the bad news is that errors in the calculated entropy are also conserved. These considerations apply to any method, even in Eulerian coordinates, in which successful measures are taken to combat numerical diffusion. Where flow features are fully resolved at the Navier-Stokes scale, there is usually no problem. The difficulty arises because we wish to avoid anomalies that are associated with unresolved features.

P.L. Roe (B) University of Michigan, Ann Arbor, MI, USA e-mail: [email protected]


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Fig. 1 In Sod’s problem, Lagrangian and shifted Eulerian methods behave identically

The simplest way consists of re-introducing diffusion, hopefully in a selective manner [4, 5]. When a Riemann solver is used to evaluate the flux, the (local) use of a more diffusive flux can be successful [2]. However, if methods are local in time, it is impossible for the code to determine, at any one timestep, whether the entropy changes being propagated are physically meaningful or due to numerical error. It follows that any method that avoids the pitfalls must either avoid making the error in the first place, or must be equipped with some kind of memory. The method of Roe and Zaide [8] seems to remove about half the error by comparing conservative and nonconservative pressure calculations (See also [3]). In this chapter, we concentrate on identifying the causes of the error. We have identified two main causes, and believe that there are no other important ones. However, we do not yet have any practical cures to propose.

2 The Noh Problem A canonical example is the Noh problem [5], a seemingly trivial Riemann problem with initial data corresponding to the collision of two equal shocks, or equivalently the reflection of a single shock from a solid wall. In Eulerian variables the data is u L = (ρ0 , ρ0 u 0 , E 0 )T , u R = (ρ0 , −ρ0 u 0 , E 0 )T or in Lagrangian variables v L = (V0 , u 0 , i 0 )T , v R = (V0 , −u 0 , i 0 )T . We take γ = 5/3, ρ0 = 1, u 0 = 1, and p0

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left to set shock speed. Virtually all shock-capturing methods of either type provide quite good solutions for pressure and velocity, but predict too small a density in a small region at the origin. In consequence the temperature there is too high, so that this and related phenomena have been called wall heating. In the more than 20 years since Noh proposed the problem, no satisfactory solution has been exhibited that would carry over to other settings, nor is there even any generally accepted explanation of the mechanism. Here we demonstrate that one driving mechanism is the one put forward by Xu and Hu [9] for anomalies associated with general shock collisions, and that the Noh problem is merely the simplest instance of a pervasive difficu lty, arising from the convexity of the pressure as a function of the conserved variables. Analysis of the first time step (which is effectively identical in both methods) is instructive. For example, any consistent finite-volume method that avoids explicit diffusion will place identical fluxes on all interfaces except for the one at x = 0, where the only non-zero flux is the interface pressure in the momentum equation. Solutions therefore depend on just two parameters, one being this pressure p, and the other being the Courant number ν, defined here as ν = SΔt/Δx with S being the speed of the reflected shock. Whatever pressure P is used, the conserved variables will change in proportion to ν. Then the pressure in the cells next to the origin will be updated using the equation of state. For example, using the exact Riemann solution for an ideal gas, this results in the following expression for the pressure in the first cells as a function of Courant number; p(ν) = νp1 + (1 − ν) p0 +

γ −1 2

ν(1 − ν)ρ0 ρ1 u 20 . (νρ1 + (1 − ν)ρ0 )

(1)

The leading terms represent the exact evolution of the pressure. The remaining term, which is always positive, reflecting the convexity of the pressure law, is an error arising from the evaluation of internal energy as the difference in total and kinetic energies. Although the error vanishes for ν equal to 0 or 1.0, it cannot be removed by choosing ν = 0 (no progress) or ν = 1.0 (unstable). At the second and subsequent time steps, this excessive pressure expels mass from the central region. The pressure and velocity reach constant values by emitting acoustic waves, but the density is left too low. This density (temperature) anomaly is traceable to initial errors in pressure, or, equivalently, entropy. This error persists until late times because there is no diffusion mechanism to remove it. Similar effects occur in methods employing artificial viscosity, since this is also a form of mixing.

3 Isolating Pressure Convexity If the explanation given above is both correct and complete, then no phenomenon analogous to wall heating should result if the pressure were to depend linearly on the conserved variables. There is no physically meaningful closure of the gasdynamic

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Fig. 2 A p − ρ plot for the Ideal Gas EOS (left) and Linear EOS (right)

equations with this property, and the only nonphysical possibility that is dimensionally consistent is to take the pressure as proportional to the total energy rather than the internal energy. In this way the pressure itself becomes a conserved variable. Therefore we set p = (γ − 1)E.

(2)

Under this assumption, the gasdynamic equations remain hyperbolic, with regular shocks, rarefactions, and contacts. We want to see if initial data analogous to the Noh problem results in an error being deposited on the wall. In Fig. 2, we plot the evolution of pressure and density in the cell next to the wall, as the solution there evolves to its steady state. In the left picture we see that with the true equation of state, either the pressure is overpredicted, or the density underpredicted, depending on the point of view. In either case the temperature is overpredicted. On the right (having contrived the data so that the final densities are the same) we see that eliminating the need for nonlinear pressure evaluation indeed eliminates the temperature error, but only for small times. This compels the conclusion that there is at least one other driving mechanism.

4 Shock Motion A clue to this comes from modifying the Noh problem by specifying the initial condition at t < 0. This is, we set up initial conditions corresponding to two shocks that will collide in the future, but have not yet done so. To our surprise, the amount of wall heating turned out to depend on the initial distance of the shock from the wall. We realised that this was related to the well-known way [1, 7] in which a propagating shock changes its structure depending on its phase relative to the grid. The shock reflects from the wall in a way that depends on its phase at the moment


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Fig. 3 (Left) Density at final time. Effects at the wall and where the contact has intercepted the reflecting shock are observed. (Right) Density contours. The propagation of the spurious contact is observed, and its residual effect remains

of impact. In some cases the outcome can even be wall cooling, but in many cases the most pronounced error was not actually at the wall. The location was at a point that can be simply predicted by accounting for the fact that a shock that is initially prescribed with too narrow an initial structure will suffer “start-up errors”. These cause spurious waves to propagate along the other fa milies of characteristics. This is often referred to as the “slowly-moving shock” problem, although it can, and does, occur for strong shocks propagating at any speed. In the example shown, a spurious contact discontinuity propagates inward with speed −u 0 . It is eventually intercepted by the reflected shock, and brought to rest. The spurious, stationary, contact becomes the location of the worst temperature error as seen in Fig. 3. Again, we examined the validity of this explanation by constructing an artificial system. Arora and Roe [1] conjectured that the spurious waves would not be produced in any system for which the Hugoniot curve is a straight line in the phase space of conserved variables. They noted, but did not show, an example using a 2×2 model system. Here we study a 3 × 3 model system, to introduce the possibility of a contact discontinuity: ⎡ ⎤ ⎡1 2 ⎤ 2 2 u 2 u +a +b ⎣a ⎦ + ⎣ ⎦ =0 ua b t ub x

(3)

√ This hyperbolic system has a propagation speed c = a 2 + b2 , eigenvalues of u, u ± c. It is simple to show that the Hugoniot curves are actually straight lines. Because there is no explicit equation of state, there is no nonlinear mixing involved. Tests on this system with left and right states of u L = (1, 1, 1)T , u R = (−1, 1, 1)T in Fig. 4 show no spurious waves, and no trace of problems such as wall heating.

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5 Conclusions and Future Work Whenever entropy is created by numerical error, it is propagated along particle paths as though it had been created physically. The simplest example of this is wall heating, exemplified by the Noh problem. Two mechanisms behind this phenomenon are examined and isolated. The explanations were validated by constructing examples of artificial conservation laws from which the mechanisms were absent, and then verifying that the phenomenon did not occur. We are currently looking at various shock-broadening mechanisms. Acknowledgements This research was supported by the DOE NNSA/ASC under the Predictive Science Academic Alliance Program by grant number DEFC52-08NA28616.

References 1. Arora, M., Roe, P.L.: On postshock oscillations due to shock capturing schemes in unsteady flows. J. Comput. Phys. 130(1), 25–40 (1997) 2. Donat, R., Marquina, A.: Capturing shock reflections: an improved flux formula. J. Comput. Phys. 42–58 (1996) 3. Fedkiw, R.P., Marquina, A., Merriman, B.: An isobaric fix for the overheating problem in multimaterial compressible flows. J. Comput. Phys., 148(2). 545–578 (1999) 4. Menikoff, R.: Errors when shock waves interact due to numerical shock width, SIAM J. Sci. Comp., 15, 1227 (1994)


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5. Noh, W.F.: Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux. J. Comput. Phys. 72, 78 (1987) 6. Rodzhestvenskii, B.I., Yanenko, N.N.: Systems of quasilinear equations. AMS Translations of Mathematical Monographs, 55 (1983) 7. Roberts, T.W.: The behavior of flux-difference schemes near slowly-moving shocks. J. Comput. Phys. 90, 141–160 (1990) 8. Roe, P.L., Zaide, Daniel, W.: An Eulerian Look at Lagrangian CFD. Numerical Methods for Multi-material Fluids and Structures Conference, Sept. 2009 9. Xu, K., Hu, J.: Projection dynamics in Godunov-type methods. J. Comput. Phys. 142, 412 (1998)

Automatic Time Step Determination for Enhancing Robustness of Implicit Computational Algorithms Chenzhou Lian, Guoping Xia, and Charles L. Merkle

Abstract A method for enhancing the robustness of implicit computational algorithms without adversely impacting their efficiency is investigated. The method requires control over two key issues: obtaining a reliable estimate of the magnitude of the solution change and defining a realistic limit for its allowable variation. The magnitude of the solution change is estimated from the calculated residual in a manner that requires negligible computational time. An upper limit on the local solution change is attained by a proper non-dimensionalization of variables in different flow regimes within a single problem or across different problems. The method precludes unphysical excursions in Newton-like iterations in highly non-linear regions where Jacobians are changing rapidly as well as non-physical results during the computation. The method is tested against a series of problems to identify its characteristics and to verify the approach. The results reveal a substantial improvement in the robustness of implicit CFD applications that enables computations starting from simple initial conditions without user intervention.

1 Introduction Computational algorithms for systems of partial differential equations traditionally use iterative methods. The efficiency and reliability represent crucial issues for practical applications. An implicit goal of any algorithm is to maximize computational efficiency while minimizing user intervention and the specification of user defined control variables at input. An ideal algorithm should converge reliably from a simple physical condition while using a standard set of predefined control variables. This ideal is far from conventional experience. Without watchful user intervention, solutions often diverge as nonlinearities cause local Jacobians to drive the solution in improper directions producing unrealistic physical variables such as negative temperatures or densities. C. Lian (B) Purdue University, West Lafayette, IN 47907, USA e-mail: [email protected]


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Although convergence acceleration has been the subject of many studies [1, 4], code robustness generally relies upon ad hoc procedures such as increasing the artificial dissipation or ramping the CFL in the early stages of the computation. In such cases, the starting and final values and the allowable rate of increase are unknown and vary dramatically among problems. Another common procedure is to reset variables such as density and temperature to small positive values when they change sign, but resetting variables does not remove the underlying cause and negative quantities generally reappear in succeeding time steps. Often these methods are combined with other similar techniques to try to improve code robustness. A limitation is that all are performed in an “open-loop” manner – the correction is devised before the problem appears, and is implemented without regard for why the difficulty appears. The goal of the present article is to investigate a “closed-loop” procedure for improving robustness that controls the local CFL during the iteration on the basis of the convergence process itself rather than by means of an a priori procedure. The method chosen is based upon a concept suggested by Luke et al. [2], but differs from their approach in that it requires negligible computational overhead and has been extended to improve its effectiveness. The objectives are to control the change of all variables at every grid point and iteration in such a manner that the same CFL can be used for all computations while simultaneously decreasing the sensitivity to initial conditions and grid distribution and ensuring that nonphysical values are not encountered. The closed-loop control is achieved by specifying a maximum solution change that allows the code to determine a local CFL in such a manner that this change is not exceeded, thereby allowing large CFL’s in non-sensitive regions of the convergence process while restricting the CFL in rapidly changing regions. We call this control method the solution-limited time-stepping method.

2 Governing Equations The Navier–Stokes equations can be written as: ∂Q +∇ ·F =0 ∂τ

(1)

where Q = (ρ, ρu, ρv, ρh 0 − p)T is the conservative variables vector, τ is a pseudo time used for iterative solution and F includes the inviscid and viscous flux vectors. For numerical solutions we choose the primitive variables, Q p = ( p, u, v, w, T )T , as the primary independent variable, so that Eq. (1) becomes: p

∂Qp +∇ ·F =0 ∂τ

(2)

The pressure and temperature in the primitive variables complement the variables in the constitutive relations while the Jacobian matrix, p , provides a transformation between Q and Q p .

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To complete the iterative procedure we replace the pseudo-time operator in Eq. (2) by an Euler implicit difference and replace the remaining terms by finite volume, discrete forms to yield the final algebraic system: p

Q k+1 − Q kp p τ

* + = − (∇ D · F )k+1

(3)

Here, the superscript k represents the running variable in pseudo time, and ∇ D is a discrete divergence operator. The equation is solved by linearizing the terms at the new pseudo-time level by Taylor’s series to give the final linear, algebraic system: *

I + τ −1 p (∇ · A p )

+k

k Q p = −τ −1 p {∇ D · F }

(4)

In this expression, the flux Jacobian is defined as A p = ∂F /∂ Q p .

3 Solution Limited Time-Stepping Method The goal of the present work is to describe a means for monitoring the solution change at every grid point and every step so that local CFL’s can be defined to maximize progress toward convergence while minimizing the likelihood of divergence. The method involves two important concepts: an efficient estimate and an acceptable magnitude of the solution change. The estimate of the solution change, Q p , as a function of the local time step size must be efficient. In principle, the time step at each grid point can be optimized by computing a series of implicit time steps, but this is intractable since the preferred time steps will be different at every grid point and the change at each grid point will be dependent on the change at neighboring cells as well as local values. Following the spirit of the method suggested by Luke et al. [2], we base the estimated solution change (which we designate as Q est p ) on the residual of the update relation in Eq. (4): k −1 } {∇ Q est = −τ · F D p p

(5)

The estimated solution change in Eq. (5) is proportional to the time step, τ . Consequently, for any cell in which Q p est is deemed large enough to cause convergence difficulties, the local time step is reduced to bring Q est p to an acceptable magnitude. This time-step-limiting procedure will be useful if Q est p constitutes a reasonable bound on Q p . The adequacy of this estimate must be verified, but as a first step, we compare Q est p and Q p by combining Eqs. (4) and (5) to obtain: * +k −1 Q p Q est p = I + τ p (∇ D · A p )

(6)

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For small τ , the second term in the brackets is negligible and the calculated and est Q est p and Q p are approximately equal, but for large τ, Q p , increases without bound (Eq. 5) whereas Q p approaches a constant. The behavior at these two limits suggests that the estimated solution change may serve as a nominal upper bound on the calculated change for all τ and that the norm of Q p may be bounded by the norm of Q est p . This supposition must however be verified. In Sect. 4, we use analytical and numerical methods to demonstrate the effectiveness of Q est p as a useful estimator of the calculated solution change, Q p . To define the allowable magnitude of the solution change we introduce a reference vector, Q pref and a scalar multiplier, α. We then limit the local pseudo-time step, τ , in such a manner that the estimated solution change, Q est p , will be less than: Q est p ≤ α Q pr e f

(7)

This limit is applied to every variable at every grid point with the local time step being chosen to satisfy the most restrictive variable in each cell. The effectiveness of this method depends upon defining appropriate values for the reference vector and the scalar multiplier. One choice for the reference vector, Q pref , is to set it equal to the current solution vector in each cell. With this reference, it is readily seen that the scalar multiplier, α, must take on two different characters depending on the nature of the variable being limited. For positive-definite quantities such as pressure, density or temperature that are not allowed to change sign, positivity is ensured by choosing α less than unity. Our choice for such quantities is α = 0.1, allowing a 10% change. Indefinite variables such as velocity, however, must be allowed to change sign and require α > 1 or a reference value larger than the local cell value. Other computations (not included) based upon used a global characteristic velocity as the reference speed with the value of α set equal to 0.1 as for a positive definite variable have also proven effective for applications in which such a velocity is easily discerned. The local thermodynamic pressure, however, cannot be used as the reference pressure, since only the pressure gradient enters the equations. For example, in incompressible flows, the thermodynamic pressure does not even enter the solution. The time-step limiting procedure can be summarized as: 1. Specify global Courant number large enough to provide rapid convergence as the solution nears the solution. For the present examples, CFLspecified = 1, 000; 2. Choose Q pref based upon local conditions as outlined above; 3. Set the maximum allowable fractional change, α; 4. Determine the estimated solution change, Q est p ; from the residual; at every grid point with the allowable solution change, α Q pref , 5. Compare Q est p and for cells that exceed the limit, decrease the local Courant number according to: CFLallowable = CFLspecified α Q pref /Q est p ; 6. Use the resulting Courant number to complete the computation.


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4 Results To demonstrate the limiting procedure, we present results for a reacting, turbulent flow in a model rocket engine combustor [3]. The combustor is fed by a coaxial injector with oxygen in the central passage and hydrogen in the surrounding annulus. The 46,000-cell computational domain is shown in Fig. 1. The combustion chamber length is 93 mm and its diameter is 38.1 mm. The Reynolds numbers of the oxidizer and fuel inlet are 604,000 and 169,000, respectively. The mass flow rates and total temperatures are specified as upstream boundary conditions and the back pressure is specified at the unchoked nozzle exit. All wall are set as no-slip, adiabatic conditions. Combustion (H2 and O2 ) is modeled by a 9 species, 17 reaction step chemical kinetics model Thermodynamic and transport properties of all species are expressed as arbitrary functions of pressure and temperature with appropriate mixing relations used to obtain mixture properties. To demonstrate the difficulty of converging this problem by conventional methods, we first attempt to solve the problem by ramping the CFL. The results of a series of three unsuccessful attempts are given in the left plot of Fig. 2 while a successful ramping procedure is given in the right side of Fig. 2 along with results for the solution-limiting procedure. The three attempts shown in the plot on the left corresponds to ramping the CFL from 0.01 to 100, 0.01 to 10 and 0.01 to 1 during the first 100 iterations. All three cases diverge before the CFL has reached its terminal value, although the ramping to CFL = 1, nearly reaches 100 iterations. As a fourth attempt, we ramp CFL from 0.01 to 1 in 500 iterations rather than 100 (right side of Fig. 2). This ramping results in a converged solution although the convergence is considerably slower than that obtained with the solution-limiting method. The results indicate that the solution-limiting procedure not only enhances code robustness without user intervention, but also improves convergence by a factor of ten. Clearly, the manually scheduled CFL ramping could be tuned to improve convergence but the uncertainties in determining the schedule and considerable user intervention that has already been done strongly contrasts with the effectiveness of the solution-limiting procedure. The calculated and estimated solution changes given in Fig. 3, again show the calculated change is well bounded by the estimated solution change.

Fig. 1 Generic geometry and computational domain. Hydrogen and oxygen enter through injector passages on left

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Fig. 2 Convergence rate comparisons for the time stepping method and traditional CFL ramping method

Fig. 3 L ∞ norm of solution-limited convergence rate for combustion in rocket engine combustor. CFL = 1, 000, α = 0.1

5 Summary and Conclusions A method for diminishing user intervention in implicit CFD computations is presented. The method is based upon controlling the local CFL in such a manner as to ensure that the local solution change is never more than a small fraction of the magnitude of the solution variable. The magnitude of the solution change is estimated from the residual requiring only a negligible increase in the per-step computational requirements. In addition to the results presented here, the method has also been tried for a large number of other problems. All solutions started from a quiescent initial condition. A single, user-specified CFL = 1, 000 was used for all cases, and all solutions converged reliably and efficiently without user intervention or changes in the control parameters apart from those variations determined by the limiting procedure. Our


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experience has been that the method greatly decreases the number of unsuccessful runs and nearly eliminates the need for user intervention in a wide variety of problems. Overall, the solution-limited time stepping procedure automatically adjusts the CFL number for each cell according to the estimated solution change and greatly reduces the sensitivity to initial conditions for various types of flows. The method is easily implemented into any implicit CFD code.

References 1. Bücker, H.M., Pollul, B., Rasch, A.: On CFL Evolution Strategies for Implicit Upwind Methods in Linearized Euler Equations. Technical Report, RWTH Aachen University, Aachen (2006) 2. Luke, E.A., Tong, X.-L., Wu, J., Tang, L., Cinnella, P.: A Step Towards “Shape-Shifting” Algorithms: Reactive Flow Simulations Using Generalized Grids. AIAA Paper No. 2001– 0897 (2001) 3. Tucker, P., Menon, S., Merkle, C., Oefelein, J., Yang, V.: An approach to improved credibility of CFD simulations for rocket injector design. In: Proceedings of 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, Jul 8–Jul 11, 2007, Cincinnati, OH (2007) 4. Vanderstraeten, D., Csk, A., Rose, D.: An Expert-system to control the CFL number of implicit upwind methods. Technical Report TM 304, Universiteit Leuven, Leuven (2000)

Part VI

Boeing–Russia Cooperation

Fifteen Years of Boeing–Russia Collaboration in CFD and Turbulence Modeling/Simulation Sergey V. Kravchenko, Philippe R. Spalart, and Mikhail Kh. Strelets

Abstract The paper highlights major outcomes of a more than 15-year tight collaboration between scientists and engineers of Boeing Commercial Airplanes (Seattle) and a group of scientists from St.-Petersburg under the aegis of the Moscow Boeing Technical Research Center. The collaborative research covers a wide spectrum of basic and applied aerodynamic problems and has resulted in numerous prominent achievements in Computational Fluid Dynamics, turbulence modeling and simulation, and Computational Aero-Acoustics (CAA). The paper focuses on the turbulence modeling and simulation aspects of the mutual work (the CAA studies are presented in a separate paper of M. Shur et al. in this volume).

1 Introduction In 1993, soon after the foundation of the Boeing Technical and Research Center in Moscow (BTRC), the first author organized a visit of a group of Boeing’s leading experts in CFD and turbulence modeling headed by Dr. W.-H Jou to Moscow, St.-Petersburg, and Novosibirsk aimed at letting them get acquainted with a work of Russian scientists in these areas. All the three authors of this paper participated in a series of meetings in Moscow which took place in the course of this visit and well remember their friendly and creative atmosphere. As a result, a number of bilateral, based on common scientific interests, groups have been formed, thus giving a start to a long-term mutually beneficial collaboration which is continuing for more than 15 years now. As far as CFD and turbulence modeling are concerned, initially this group included several scientists from the State Institute of Applied Chemistry (SIACh) and St.-Petersburg State Polytechnic University (SPbSPU) headed by Prof. Strelets and from the Central Institute of Aviation Motors Building in Moscow headed by S.V. Kravchenko (B) The Boeing Company, Chicago, IL 60606-1596, USA e-mail: [email protected]


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Prof. Secundov, from one side, and Dr. Spalart and some other scientists and engineers from Boeing Commercial Airplanes (BCA) in Seattle, from the other side. Later on the major part of the work in these areas moved to St.-Petersburg and was carried out in the framework of a long-term Collaboration Agreement between BCA, SIACh and R & D company “New Technologies and Services” (NTS) under general supervision of P. Spalart, the Technical Projects monitor at BCA. Starting from then, a core of the team remains essentially unchanged, which in itself is far from typical for international R & D activity. A major reason of this “stability” and a high productivity of the joint team is that the relationships between Seattle and St.-Petersburg are not customer-supplier-like, which presumes formulating SOW’s from one side and delivering reports from the other side. They are rather joint-team relationships, with a daily communication between the researches and engineers from both sides. This permits an efficient combination of not only scientific and technical expertise but also cultural backgrounds of both groups and makes their mutual work a great intellectual and human adventure, bridging the two continents. An outcome of the 15 years work, which is outlined in 32 volumes of multi-pages reports, 43 papers in the archived US, Russian, and European Journals and in proceedings of 54 International and National Conferences and Workshops, cannot be, of course, presented in one publication with any reasonable level of completeness. So in this paper we just highlight a few achievements reflecting different aspects of the work which includes a basic research in the area of turbulence modeling and simulation and applied studies directed to solution of specific practical problems. However before doing this we briefly outline the NTS code which is an in-house CFD code developed in the course of the project and used in all the studies which results are presented below.

2 NTS CFD Code This code was designed in St.-Petersburg in the middle of the nineties based on the expertise of the CFD group of SIACh. It is a finite volume code which accepts 2D and 3D structured multi-block grids of the Chimera type and computes compressible (at arbitrary Mach number) and incompressible, steady and unsteady flows within different approaches to turbulence treatment (RANS, DES, SAS, LES, and DNS). A range of numerical schemes implemented in the code includes implicit high order (3rd or 5th) upwind and hybrid (5th order upwind/4th order centered) flux difference splitting scheme of Rogers and Kwak for incompressible flows and of Roe, Van Leer, and Weiss and Smith schemes for compressible flows. Numerical implementation of these schemes is performed by implicit relaxation algorithms (Plane/Line Gauss-Seidel relaxation, LU relaxation, and DDADI algorithm), which may be arbitrarily specified by a user in different grid-blocks. During 15 years passed from the first release of the NTS code aimed at computations on

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one-processor PC’s, it has undergone significant enhancements including adding massively parallel capability and can now run on contemporary PC’s clusters and mainframe super-computers. The code has passed extensive code-to-code comparisons with other public, in-house industrial, and commercial CFD codes (CFL3D of NASA, CGNS of Boeing, ELAN of the Technical University of Berlin, CFX and FLUENT) and, as of today, is considered as one of the most reliable and efficient CFD codes for aerodynamic applications. Thanks to this, even with a rather restricted CPU power available to the team, especially in the early stages of the work, it has been always competitive with and often superior to other groups possessing much more powerful computers. As an example, Fig. 1 presents flow visualization from DNS of the flat plate turbulent boundary layer at the momentumthickness Reynolds number 666 with wall-mounted Large-Eddy Break Up (LEBU) device carried out on PC with a grid of about 3 million cells [11].

Fig. 1 Fragment of the computational grid and instantaneous vorticity contours from DNS of turbulent boundary layer with LEBU [11] (flow from left to right)

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3 RANS Turbulence Modeling RANS studies carried out by the team include validation of existing and newly developed/enhanced RANS models and numerous computations of aerodynamic flows of practical interest. A space limit of the paper does not permit to dwell upon these studies in any detail, but two proposals of the team illustrating its activity in this area still deserve mentioning. The first one is a general approach to sensitization of eddy-viscosity turbulence models to rotation and curvature and a Rotation-Curvature correction to Spalart-Allmaras one-equation eddy viscosity RANS model (SARC) based on this approach. Proposed in the work of Spalart and Shur [10] in 1996 it still remains a leader in this field and is superior even to Reynolds Stress Transport models which, in principle, should account for the RC effects “by definition”. The model is simple and robust and has been proven to provide accurate predictions of a wide range of aerodynamic and industrial flows of great practical importance in which other available RANS models turn out to be helpless. An example of SRAC performance in the vortical wake of an airplane at high lift configuration [5] is presented in Fig. 2 demonstrating an excellent agreement of CFD with experimental data. One more example is CFD-based design and optimization of the Vortex Generators for 737 flight-deck noise reduction carried out with the use of the SARC model in [1] (see Fig. 3). The second of the abovementioned proposals is an original approach to modeling of separation of laminar boundary layers from bluff-bodies and transition to turbulence in the separated shear layers in the framework of RANS [2]. Later on it has

Fig. 2 Comparison of computed wake vortex evolution with experiment [5]


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Fig. 3 Photographs of VG’s installed on airplane and 1/3-octave band sound pressure level measured in flight test at 35,000 feet and Mach number 0.78 at pilot’s outboard ear with and without VG’s

been proven to be compatible with hybrid RANS-LES turbulence models and now serves as an efficient tool for computing sub-critical flows over bluff bodies with the use of such models.

4 Detached-Eddy Simulation of Turbulence This is may be the most prominent achievement of the joint team in the area of turbulence modeling. Proposed in 1997 [9], Detached-Eddy Simulation (DES) turned the “vector of efforts” of turbulence research to hybrid RANS-LES approaches, which made simulations of extremely complex massively separated flow possible even on PC’s and are currently widely used all over the world. A basic idea of DES is using a unique turbulence model which automatically functions as a RANS model in the whole attached boundary layer and as a sub-grid scale LES model in separation regions away from the solid walls, thus seamlessly coupling both approaches in a way permitting to employ their best features, namely, the computational efficiency and accuracy of RANS in attached boundary layers and universality and affordable CPU of LES in the separation regions. A success of the first applications of DES to flows past airfoil at high (beyond stall) angles of attack

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[3] and to sub- and super-critical flows past circular cylinder [12], resulted in its rapid dissemination (currently DES is implemented in many academic and most of the industrial and commercial CFD codes) and extensive use.1 This, in turn, has permitted to highlight some issues with the original DES approach, analyzed in detail in a lecture given by Dr. Spalart at ICCFD-3 in 2004 [6]. These issues were addressed in further works of the Boeing-NTS team which resulted in a modified version of DES (Delayed DES or DDES [8]) resolving the issue of Grid-Induced Separation caused by a premature switch from RANS to LES mode within attached boundary layer, the original version of DES suffers from in case of the so-called “ambiguous” grids [6]. Then DDES has been further improved by adding the Wall Modeled LES capability (Improved DDES [4]), which has not been originally viewed as a natural DES application area. A detailed analysis of the current status of DES and its modifications and enhancements is given in the review paper of Spalart [7]. Note that only a list of successful simulations carried out with the use of DES, DDES, and IDDES by their authors and other researches and engineers all over the world would occupy several pages. So here we present only two examples of such simulations. The first one is a kind of “exotic”: it is DES of the flow over raised airport runway of the Santa Catarina Airport (Funchal, Madeira Island, Portugal) which photograph in the course of construction is shown in Fig. 4. The runway was built on stilts, 185 m wide and 58 m high on the island-side. This rare configuration, in case of cross-wind from the ocean, was likely to cause separation off the edge of the runway platform, creating lateral wind shear at some height over the runway and reversed flow at its surface, as well as high relative turbulence intensity, and the objective of DES study carried out in the course of the runway building was to quantify these effects. The flow visualization of DES carried out on a modest grid of 750,000 cells reveals separation of the flow off the edge of that runway platform, shear layer roll-up, and formation of essentially 3D roller/rib structures, as is typical for thin bodies at high angles of attack. In order to assess the runway flow conditions

Fig. 4 Photograph of runway and visualization of wind-flow over the runway

1 This has been to a considerable extent promoted by participation of SPbSPU and NTS in the European projects FLOMANIA, DESider, and ATAAC devoted to development and validation of new turbulence modeling and simulation approaches.


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quantitatively minima and maxima of the x-velocity component were collected from the simulation over time and the spanwise domain. These data have shown that the range of the velocity variations (the strength of the “wind gusts”) over the runway may reach ±100% of the nominal wind velocity. This information has been used by the airport administration for defining maximum wind velocity for safe operation of the runway. The second example is a very recent application of DDES for the tandem cylinders flow which is of both significant academic and practical interest as a prototype for interaction problems commonly encountered in airframe noise configurations, e.g., a landing gear (the simulation was performed with the use of resources of the Leadership Computing Facility at Argonne National Laboratory). Figure 5 shows the flow visualization, which reveals extremely complex fine turbulence structures resolved by the simulation both in the gap between the cylinders and in the rear cylinder wake and reflects the crucial growth of the computer power compared to the 2000 year simulation shown in Fig. 4. Note however that even with the large grid (about 60 million points) and low-dissipative numerics used in the simulation, the agreement with the experiment of some of the flow quantities remains far from perfect, thus suggesting that far not all the issues associated with a reliable prediction of the considered flow are successfully resolved. First of all, this is related to the accurate representation of the evolution of separated shear layers which remains a great physical and computational challenge.

Fig. 5 Visualization of tandem cylinders flow (λ2 isosurface)

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References 1. Anderson, B., Shur, M., Spalart, P.R., Strelets, M., Travin, A.: AIAA Paper, 2005-0426 (2005) 2. Shur, M., Spalart, P.R., Strelets, M., Travin, A.: In: Proceedings of the 3rd ECCOMAS CFD Conference, pp. 676–682 (1996) 3. Shur, M., Spalart, P.R., Strelets, M., Travin, A.: In: Proceedings of ETMM 4, 669–678, Elsevier, Amsterdam, Lausanne, New York, Shannon, Singapore, Tokyo (1999) 4. Shur, M., Spalart, P.R., Strelets, M., Travin, A.: A hybrid RANS-LES approach with delayedDES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow. 29, 1638–1649 (2008) 5. Slotnik, J., Czech, M., Yadlin, Y.: Application of OVERFLOW to the evolution of aircraft wake vortices. In: Proceedings of the 9th Overset Composite Grid and Solution Technology Symposium, Pennsylvania State University, State College, PA (2008) 6. Spalart, P.R.: Computational Fluid Dynamics 2004, 3–12, Springer-Verlag, Berlin, Heidelberg (2006) 7. Spalart, P.R.: Detached-eddy simulation. Ann. Rev. Fluid Mech. 41, 181–202 (2009) 8. Spalart, P.R., Deck, S., Shur, M.L., Squires, K.D., Strelets, M.Kh., Travin, A.K.: A new version of detached-eddy simulation, resistant to ambiguous grid densities. Theor. Comp. Fluid Dyn. 20, 181–195 (2006) 9. Spalart, P.R., Jou, W.-H., Strelets, M., Allmaras, S.R.: Advances in DNS/LES. In: Proceedings of the First AFOSR International Conference on DNS/LES, Greyden Press, Columbus, OH (1997) 10. Spalart, P.R., Shur, M.: On the sensitization of turbulence models to rotation and curvature. La Recherche Aerospatiale 3, 465–474 (1996) 11. Spalart, P.R., Strelets, M., Travin, A.: Direct numerical simulation of large-eddy-break-up devices in a boundary layer. Int. J. Heat Fluid Flow. 27, 902-910 (2006) 12. Travin, A., Shur, M., Strelets, M., Spalart, P.R.: Detached-eddy simulations past a circular cylinder. FTAC 63, 293–313 (1999)

LES-Based Numerical System for Noise Prediction in Complex Jets Mikhail L. Shur, Andrey V. Garbaruk, Sergey V. Kravchenko, Philippe R. Spalart, and Mikhail Kh. Strelets

Abstract An overview is presented of a non-empirical CFD/CAA numerical system for jet noise prediction developed by the joint US – Russia team since 2002. Key elements of the system are discussed and examples are considered of its application to flow and noise computation for a wide range of cases that progress in the direction of the complete simulation of exhaust jets from real airliner engines.

1 Introduction Reductions of the noise generated by turbulent exhaust jets behind aviation engines are of great practical importance since this type of aerodynamic noise is the main contributor to aircraft noise at take-off. Not surprisingly, during the last decade intense efforts have been invested in a search for efficient improvements in this area. As a result, a number of technical devices have been proposed and tested, but as of today the “target” value of external aircraft noise reduction by 10 EPNdB without excessive penalties is still far from reached. To some extent, this is caused by the absence of a reliable computational tool for jet noise prediction. This work presents the status of such a tool, developed in the course of a longstanding collaboration between the scientists and engineers from the Boeing Company on one side and the R & D Company “New Technologies and Services” and St.-Petersburg State Polytechnic University, on the other side. This is an LES-based CFD/CAA numerical system which is ultimately aimed at predicting the aerodynamic characteristics and the noise of jets of real airliner engines with engineering accuracy of 2–3 dB, while having no empiricism and a general-geometry capability. The problem of LES-based prediction of the noise generated by turbulent jets presents a great challenge, in terms of both numerics and physics, primarily because of the need to resolve multiple turbulent and sound scales and, also, because of the complexity of combining turbulence and far-field acoustics. The difficulties are M.L. Shur (B) New Technologies and Services and State Polytechnic University, St. Petersburg 195220, Russia e-mail: [email protected] A. Kuzmin (ed.), Computational Fluid Dynamics 2010, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-17884-9_18,

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aggravated by many non-trivial flow features, such as complex aerodynamics of the jets from non-circular and dual-stream nozzles, strong temperature variations, the presence of shocks (in case of imperfectly expanded supersonic jets), etc. These highlight the importance of a number of “strategic” decisions that are needed for a successful noise computation. They include the choice of the numerical scheme; the configuration of the computational domain, grid topology, and boundary conditions; the Subgrid-Scale (SGS) model (if any); the approach to reproducing transition to turbulence, etc. The same is true regarding the choice of an optimal method of farfield noise extraction from an LES in a confined computational domain. The paper outlines the most important of these strategic decisions made in the numerical system we have developed and presents a range of “academic” and applied jet-noise studies demonstrating its current capabilities.

2 Overview of the Developed Numerical System A detailed description of the numerical approach the system is based on is given in [2, 6, 8], and here we only briefly outline its salient features. As far as numerics are concerned, the system is implemented within a generalpurpose structured multi-block massively parallel Navier – Stokes solver (NTS code [7]), which uses implicit 2nd order time integration and dual time stepping. The approximation of the inviscid fluxes is based on the flux-difference splitting method of Roe and employs a weighted (4th-order centered and 5th-order upwind-biased) scheme in the turbulent region and acoustic near field coupled with a “pure” upwindbiased one outside that region. The weight of upwinding is chosen to keep numerical dissipation at the lowest level sufficient to prevent numerical instabilities introduced by nonlinearities, grid stretching, and other sources. A major element of the algorithm used for treatment of jets with shocks is a simple and robust method of automatic local activation of flux-limiters which, to a considerable extent, permitted to reconcile the contradictory demands of shock capturing and resolution of fine-grained turbulence in LES and for the first time allowed satisfactory predictions of the spectral and integral characteristics of the far-field noise of under-expanded jets in a wide range of pressure ratios [3, 5, 12]. The grids used in the simulations have two overlapping blocks (additional artificial blocks are introduced to better make use of parallel processors). The inner, Cartesian, block is introduced to avoid a singularity at the axis of the cylindrical coordinates, and the outer, O-type, block allows a good control of the grid density and, in particular, grid-clustering within the thin shear layer, the latter being of crucial importance for representation of the fine-scale turbulence and therefore the high-frequency part of noise spectrum. One of the key elements of the system is an original two-stage simulation procedure with a coupled nozzle/jet plume RANS computation, in the first stage, and LES of the jet plume alone, in the second stage. The approach has proven to reproduce the effect of the internal nozzle geometry and maintain realistic boundary layers without the extreme cost of the full coupled nozzle-plume LES [5].

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For the turbulence simulation in the second (LES) stage of the computations, our current choice is to de-activate the Subgrid-Scale (SGS) model and to rely on the subtle numerical dissipation of the slightly upwind scheme, a strategy which is compatible with the spirit of LES, away from walls. Although an approximation, this approach ensures a rapid transition to turbulence in the jet shear layers, which is a key prerequisite for an accurate noise prediction at realistic (i.e., high) Reynolds numbers. A rigorous approach would be resolving the fine-scale turbulent structures of the nozzle boundary layers that seed the shear layer, but at the practical Re values this is far out of reach even with the most powerful modern supercomputers (see, e.g., [9]). An alternative approximate approach (explicit LES combined with artificial inflow forcing, as employed in many other jet studies) was rejected to avoid the creation of parasitic noise and especially the introduction of a number of arbitrary parameters. Note that our results without forcing certainly benefit from having thin boundary layers at the high Re, which reduces the extent of the unrealistic transition region, and the azimuthal correlation scale. Finally, for the far-field noise extraction, Lighthill’s acoustic analogy in the form of the permeable Ffowcs-Williams and Hawkings surface integral method is used. In contrast to the Kirchhoff approach, which could be the other practical option, it allows the placement of the control surface in the immediate vicinity of the turbulent region (in the inviscid but non-linear near-field) and, therefore, the confinement of the fine-grid area needed for turbulence resolution exactly to this turbulent area thus minimizing the loss of quality of the waves before they reach the surface, particularly for the higher-frequency waves in the near-nozzle area. The control FWH surfaces have a shape of a tapered funnel with a “closing disk” at the downstream end which turbulence necessarily crosses, in violation of the assumptions of the quadrupole-less FWH approach. The inaccuracy caused by this violation is drastically reduced by a change of variables [2, 8] proved to be much more efficient than simply omitting the disk from the integral (“opening” the control surface) as is done in many FWH-based jet noise studies.

3 Current System Capabilities The numerical system briefly outlined above has now reached a good level of confidence over a wide range of both flow conditions and geometries, as demonstrated by numerous examples of jets aerodynamics/turbulence and noise computations presented in [2–6, 12, 13]. In terms of flow conditions, the system has been shown capable of predicting the noise of subsonic and supersonic exhaust jets in the full temperature range of interest for commercial aviation. As applied to simple round jets, this is illustrated by Figs. 1 and 2, which present results of the computations [4, 12] for the cold subsonic M = 0.9 and hot sonic strongly under-expanded jets, respectively. Particularly, Fig. 1 demonstrates a clear trend to grid-convergence of the predicted noise spectra and a distinct improvement of their agreement with

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Fig. 1 Effect of grid on 1/3-octave noise spectra of cold M = 0.9 jet (data from [10])

Fig. 2 Instantaneous density gradient field (“numerical Schlieren”) and narrow-band noise spectra of a hot under-expanded sonic jet (data from [10])

measurements when the computational grid is refined. On the finest grid with around 23 million cells quite affordable nowadays, the approach ensures a reliable resolution of the spectra up to diameter-based Strouhal numbers of about 12 (compared to St = 2 on the coarsest grid with 1.7 million cells), which is already not crucially far from the range of St = 15–20 required by practice. This supports physical plausibility and numerical efficiency of the proposed simulation strategy and suggests that for the practically meaningful Re numbers and frequencies, the fully coupled nozzle-plume LES capable of “creation” of LES-content in the incoming boundary layer but unaffordable in terms of required computational resources is not really indispensable. For the shocked jet shown in Fig. 2, one can see that the spectral characteristics of the noise including its broadband shock-associated component are predicted fairly well, thus providing an indirect evidence that essential features of shock-turbulence interaction (illustrated by the upper frame in the figure) are captured in the simulation. In terms of geometries of exhaust systems that can be treated in the framework of the developed numerical tool, they now include dual nozzles, with stagger and with an external core plug, i.e. virtually fully reproduce the real exhaust jets from

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Fig. 3 Instantaneous vorticity in the meridian plane and overall sound directivity of dual-jet exhaust system with extended center body (data from [1])

Fig. 4 Effect of core nozzle exit beveling on instantaneous vorticity and overall sound directivity of dual jet in M = 0.2 external stream (data from [11])

modern turbofan aviation engines (this is not to imply that full industrial cases, meaning an installed engine with pylon, wing and flaps, have been treated yet). Different noise-reduction devices such as fan-flow deflecting vanes, chevrons, beveled nozzles, and micro-jets injected into the main jet are also routinely treated [4–6, 13]. Some examples of prediction of flow and noise for such “complex” jets are shown in Figs. 3 and 4. As seen in the figures, the accuracy of the noise prediction even with relatively coarse grids (up to 5 million nodes) is close to the “target” accuracy of 2–3 dB for both the integral jet noise and its spectral characteristics (not shown) up to a diameter-based Strouhal number ranging from 3 to 5, depending on the jet’s parameters. Thus it can be concluded that at this stage, the challenges associated with the reliable jet noise prediction appear to have been largely mastered, and the CPU power to be the essential obstacle to unrestricted performance. Remaining obstacles and areas for sustained attention in further work are: better representation of transition to turbulence in the jets’ shear-layers within the two-stage RANS-LES computational approach and addressing the full geometry complexity of industrial flows, which have a number of additional geometry features (pylons, heat shields, vents, etc.).

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References 1. Callender, B., Gutmark, E., Martens, S.: Far-field acoustic investigation into chevron nozzle mechanisms and trends. AIAA J. 43, 87–95 (2005) 2. Shur, M.L., Spalart, P.R., Strelets, M.Kh.: Noise prediction for increasingly complex jets. Part I: Methods and tests. Part II: Applications. Int. J. Aeroacoust. 4, 213–266 (2005) 3. Shur, M.L., Spalart, P.R., Strelets, M.Kh.: LES-based noise prediction for shocked jets in static and flight conditions. AIAA Paper-2010-3840 (2010) 4. Shur, M.L., Spalart, P.R., Strelets, M.Kh.: LES-based evaluation of a microjet noise reduction concept in static and flight conditions. In: Proceedings of IUTAM Symposium on Computational Aero-Acoustics for Aircraft Noise Prediction, Southampton, UK http://www. southampton.ac.uk/~gabard/IUTAM/programme.html (2010). Accessed 4 July 2010 5. Shur, M.L., Spalart, P.R., Strelets, M.Kh., Garbaruk, A.V.: Further steps in LES-based noise prediction for complex jets. AIAA Paper-2006-0485 (2006) 6. Shur, M.L., Spalart, P.R., Strelets, M.Kh., Garbaruk, A.V.: Analysis of Jet-noise-reduction concepts by Large-Eddy simulation. Int. J. Aeroacoust. 6, 243–285 (2007) 7. Shur, M., Strelets, M.Kh, Travin, A.: High-order implicit Multi-block Navier–Stokes Code: Ten-Years experience of application to RANS/DES/LES/DNS of turbulent flows. Invited lecture. 7th Symposium on Overset Composite Grids and Solution Technology, October 5–7, 2004, Huntington Beach, CA (2004) 8. Spalart, P.R., Shur, M.L.: Variants of the Ffowcs Williams-Hawkings equation and their coupling with simulations of hot jets. Int. J. Aeroacoust. 8, 477–492 (2009) 9. Uzun, A., Hussaini, M.Y.: Simulation of noise generation in near-nozzle region of a chevron nozzle jet. AIAA J. 47, 1793–1810 (2009) 10. Viswanathan, K.: Aeroacoustics of hot jets. J. Fluid Mech. 516, 39–82 (2004) 11. Viswanathan, K.: An elegant concept for reduction of jet noise from turbofan engines. J. Aircraft. 43, 616–626 (2006) 12. Viswanathan, K., Shur, M.L., Spalart, P.R., Strelets, M.Kh.: Comparisons between experiment and Large-Eddy simulation for jet noise. AIAA J. 45, 1952–1966 (2007) 13. Viswanathan, K., Shur, M.L., Spalart, P.R., Strelets, M.Kh.: Flow and noise predictions for single and dual-stream beveled nozzles. AIAA J. 46, 601–626 (2008)

Part VII

Discontinuous Galerkin Methods

Local Time-Stepping for Explicit Discontinuous Galerkin Schemes Gregor Gassner, Michael Dumbser, Florian Hindenlang, and Claus-Dieter Munz

Abstract A class of explicit discontinuous Galerkin schemes is described which time approximation is based on a predictor corrector formulation. The approximation at the new time level is obtained in one step with use of the information from the direct neighbors only. This allows to introduce a local time-stepping for unsteady simulations with the property that every grid cell runs with its own optimal time step.

1 Introduction The time discretization in discontinuous Galerkin schemes for advection diffusion reaction equations is often based on the so called ODE (Ordinary Differential Equation) or method of lines approach. In this approach the discretization in space is applied first and then the time discretization is applied in a second step using an ODE solver. The attractivity is that the space and time discretization separates which simplifies the structure and gives latitude to change time and space approximation independently. In this chapter, we describe an explicit time approximation which may be considered as a formulation in the space-time domain rather than in a separate step. The numerical scheme is kept explicit by a predictor corrector approach. Such an approach was first proposed within the finite volume framework. The second order accurate MUSCL scheme has this form which was generalized by Harten et al. [4] in their famous paper about ENO schemes. Here, a truncated Taylor expansion in time is used to predict the time evolution of the data within the grid cell. The time derivatives are successively replaced by space derivatives using the evolution equation, usually called the Cauchy Kovalewskaya procedure.

G. Gassner (B) Institute of Aerodynamics and Gas Dynamics, Universität Stuttgart, D70550 Stuttgart, Germany e-mail: [email protected]


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We generalize this procedure in this chapter and incorporate it into the DG schemes. The Taylor expansion may be considered as a local predictor which approximates the time evolution within the grid cell. Lörcher et al. [5] proposed to use a space-time expansion in the barycenter, applying the Cauchy Kovalewskaya procedure, and used this local solution to evaluate the space-time integrals in the fully discrete DG formulation. This Taylor expansion may be replaced by any other approximate solution of the local Cauchy problem which gives a space-time approximation within the grid cell. Dumbser et al. [2, 3] proposed to use a locally implicit either continuous or discontinuous Galerkin time discretization to define an auxiliary solution. Here, we describe a novel idea using a local continuous extension Runge Kutta scheme. This first step can be considered as a predictor that gives the needed time accurate data for the evaluation of the integrals in the discrete variational formulation. As every explicit scheme the predictor corrector approach has a time step restriction. For unsteady solutions this time step restriction is also a natural condition of consistency, because it guarantees the appropriate resolution of the solution in time, too. But it becomes cumbersome for unstructured grids with small grid cells to resolve the geometry or for solutions with different local time scales. Using a global time step, the grid cell with the smallest time step defines the time step for all grid cells. In our explicit DG approach, there is a remedy for this drop in efficiency. The locality of the explicit space-time DG scheme allows a completely new time marching technique: Each grid cell may run with its own time step in a time-consistent manner, thus local time stepping for unsteady problems.

2 General Formulation In the following we discuss the discontinuous Galerkin method. To keep matters simple, we restrict the discussion to a scalar conservation law of the form u t + ∇ · f = 0,

(1)

with appropriate initial and boundary conditions in a domain Ω×[0; T ] ⊆ R d ×R0+ . The flux function f (u, ∇u) is composed of two parts f = f (u, ∇u) = fa (u) − fv (u, ∇u) ,

fv (u, ∇u) = μ(u)∇u.

(2)

The first step of our approximation is to subdivide the domain Ω in non-overlapping grid cells Q. For every grid cell, we use a local polynomial approximation of the form N Q Q u(x, t) Q ≈ u Q (x, t) = uˆ j (t)ϕ j (x) =: uˆ Q (t) · ϕ Q (x), j=1

N=

( p + d)! , (3) p!d!

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* + Q where ϕ j (x)

j=1,...,N

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is a set of modal hierarchical orthonormal basis functions.

The dimension of this space N and thus the number of time dependent degrees of freedom uˆ Qj (t) depends on the polynomial degree p and the spatial dimension d. The next step of our approximation is to define how the degrees of freedom uˆ Qj (t) are determined. The base of the considered discontinuous Galerkin method is a weak formulation. We insert the approximate solution (3) into the conservation law (1), multiply with a test function φ = φ(x) and integrate over the volume of grid cell Q. We obtain u t + ∇ · f, φQ = 0.

(4)

We proceed with an integration by parts u t , φQ + f · n, φ∂ Q − fa − μ∇u, ∇φ Q = 0,

(5)

where n denotes the outward pointing normal vector and . . .∂ Q the integral over the grid cell surface. As the approximative solution is in general discontinuous across grid cell interfaces, we need an approximation by numerical flux functions. For advection as well as diffusion we apply approximate Riemann solvers. While this is standard for advection, for diffusion it is described in [6] in detail. The semi-discrete version with discretization in space only we rewrite in a compact form as

+ Q uˆ t = RV uˆ Q , ϕ Q + R S uˆ Q , uˆ Q , ϕ Q ,

(6)

where we collect all volume terms in RV and all surface terms in R S . We indicate + the dependence of the surface term on neighbor data by uˆ Q . The set of ODE’s (6) can now be integrated, where the time interval [0; T ] is subdivided into time levels tn , by using for instance the standard Runge-Kutta methods, resulting in the classic Runge-Kutta discontinuous Galerkin method, see, e.g., [1]. In this chapter a predictor corrector formulation is presented which picks up again the space-time nature of the equations. We start with an integration in time of the semi-discrete formulation (6) from time level tn to time level tn+1

Q uˆ n+1

− uˆ Qn

tn+1

+ = RV uˆ Q , ϕ Q + R S uˆ Q , uˆ Q , ϕ Q dt.

(7)

tn

The most efficient way to approximate the time integrals is to use Gauss quadrature. The problem is, that the DG solution is only known at the “old” time level tn . However, the Gauss points live in-between the time levels tn and tn+1 and thus an implicit treatment or a predictor is needed for the evaluation of the volume and surface terms. A predictor is found by approximating the following local Cauchy problems: Find

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for every grid cell Q the function v = v(x, t) for (x, t) ∈ R d × [0; t], which satisfies the initial value problem v(x, t = 0) = u ∗ (x, tn ),

vt + ∇ · f (v, ∇v) = 0,

(8)

where u ∗ (x, tn ) is the DG polynomial u Q (x, tn ) of grid cell Q extended in R d . The idea is to solve this locally in every grid cell in a first step by any numerical method that produces a space-time solution of the desired order of accuracy.

2.1 The Predictor-Corrector Formulation We propose in this chapter to apply an explicit local Runge-Kutta Galerkin discretization to construct an approximative solution to the local Cauchy problems. Accordingly to the semi discrete DG scheme described above we introduce an approximation with the same polynomial degree v Q (x, t) =

N

Q

Q

vˆ j (t)ϕ j (x) =: vˆ Q (t) · ϕ Q (x).

(9)

j=1

Inserting this into (8), multiplying by a test function and integrating over the grid cell Q yields the semi-discrete Galerkin formulation

(v Q )t + ∇ · f v Q , ∇v Q , φ Q = 0,

(10)

and analogously the set of ODE’s for the time dependent polynomial coefficients

vˆ Q = RV vˆ Q , ϕ Q , t

vˆ Q (0) = uˆ Q (tn ).

(11)

We note that the local problem (8) does not involve DG data from neighbor grid cells. As stated above we aim to use a Runge-Kutta method to integrate (11) in time. However, to evaluate the space-time integrals in Eq. (7), a continuous approximation in time is needed. In [7, 8] a special Runge-Kutta based framework for the solution of such initial value problems was introduced, with the main feature that the approximation can be naturally extended to a time polynomial, hence the name continuous extension Runge-Kutta (CERK) schemes. We observed, that for a desired time order Ot of the final scheme, we need one order less for the construction of the approximation of the local Cauchy problem Ot∗ = Ot − 1. The evaluation of (7) with the approximation v Q uˆ Qn+1

Δt − uˆ Qn

= 0

+ RV vˆ Q (t), ϕ Q + R S vˆ Q (t), vˆ Q (t), ϕ Q dt,

(12)


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increases the time order Ot by 1. Summing up, we have shown how to use a RungeKutta method to construct a time continuous local solution and insert this into the fully discrete DG scheme. If we recall the semi-discrete Galerkin formulation of the local Cauchy problem (10), we notice that the volume integral is directly related to the time derivative of the auxiliary solution

(v Q )t , φ Q = − ∇ · f v Q , ∇v Q , φ Q ,

vˆ Q = RV vˆ Q , ϕ Q . t (13)

or

Inserting this into the volume term yields tn+1 Δt t

Q Q Q Q Q RV uˆ , ϕ dt ≈ RV vˆ , ϕ dt = vˆ t dt = vˆ Q (Δt) − vˆ Q (0). tn

0

0

(14) Due to the construction of the auxiliary solution we have vˆ Q (0) = uˆ Qn . The strong variant of the fully discrete DG scheme (12) yields the predictorcorrector formulation

Q uˆ n+1

t Q

= vˆ (t) +

+ R S vˆ Q (t), vˆ Q (t), ϕ dt.

(15)

0

This formulation shows, that the DG solution at the new time level uˆ n+1 is determined by the value of the prediction at the new time level vˆ Q (Δt) (note that the predictor does not take any neighbor data into account) corrected with the surface integral term, where information from the local and the neighbor grid cells is taken into account.

3 Local Time-Stepping The most important property of this predictor-corrector formulation is its inherent locality valid for the whole time step. The disadvantage of an explicit time approximation, to advance with the smallest time step determined by all local stability restrictions, can be dropped by a time accurate local time stepping. The time evolution is shown in Fig. 1. A grid cell Q i is evolved from tin to tin+1 = tin + Δti under the condition that the predictor (light gray) of the neighbor cells Q j is available tin+1 ≤ t n+1 . This evolve condition guarantees the time-consistency needed j for unsteady problems. In Fig. 1a, all predictors are calculated and Q 2 fulfills the evolve condition. In Fig. 1b, the corrector, consisting of the surface integrals, is then applied to Q 2 , a new predictor can be computed and now Q 3 can evolve. The

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(a)

(b)

(c)

(d)

Fig. 1 Time evolution of the local time stepping algorithm in 1D with 4 grid cells

algorithm continues analogously, see Fig. 1c, d. If we take a careful look at the corrector, the spatial integral can be approximated with Gauss integration, yielding to the general form n+1

ti

n+1

, h(x, t)ϕ j (x) ds dt =

tin

∂ Qi

ti M tin

=

tin+1

j h˜ k (t) ωk

k=1

M

dt =

M k=1

j

Hk ωk ,

˜h k (t) dt ω j k

tin

(16)

k=1

where h(x, t) is the numerical flux, depending on the local and neighbor predictor, j h˜ k (t) = h(ξk , t) its value at the spatial point. The weights ωk contain the evaluation of ϕ j (ξk ) and are calculated once at the beginning of the calculation, since they are time-independent. Furthermore, integration can be changed. Now

and summation n+1 n only the time integrated flux Hk = H ξk , ti , ti at evaluation point ξk remains to be calculated. We will first integrate the fluxes in time and then in space. The remedy can be seen in Fig.1c at the interface Q 2 , Q 3 . The time integrated flux t20 , t21

is temporarily saved. We only add H ξk , t21 , t31 , before applying the final space integration. This is crucial for the efficiency of the local time-stepping algorithm.


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References 1. Cockburn, B., Shu, C.-W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. J. Comput. Phys. 141, 199–224 (1998) 2. Dumbser, M., Balsara, D. S., Toro, E. F., Munz, C.-D.: A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes on unstructured meshes. J. Comput. Phys. 227, 8209–8253 (2008) 3. Dumbser, M., Enaux, C., Toro, E. F.: Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008) 4. Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, iii. J. Comput. Phys. 71(2), 231–303 (1987) 5. Lörcher, F., Gassner, G., Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion. I. Inviscid compressible flow in one space dimension. J. Sci. Comp. 32(2), 175–199 (2007) 6. Lörcher, F., Gassner, G., Munz, C.-D.: An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227(11), 5649–5670 (2008) 7. Owren, B., Zennaro, M.: Order barriers for continuous explicit Runge-Kutta methods. Math. Comp. 56, 645–661 (1991) 8. Owren, B., Zennaro, M.: Derivation of efficient continuous explicit Runge-Kutta methods. SIAM J. Sci. Stat. Comput. 13, 1488–1501 (1992)

Multi-dimensional Limiting Process for Discontinuous Galerkin Methods on Unstructured Grids Jin Seok Park and Chongam Kim

Abstract The present chapter deals with the continuous work of extending the multi-dimensional limiting process (MLP), which has been quite successful in finite volume methods (FVM), into discontinuous Galerkin (DG) methods. Based on successful analyses and implementations of the MLP slope limiting in FVM, MLP is applicable into DG framework with the MLP-based troubled-cell marker and the MLP slope limiter. Through several test cases, it is observed that the newly developed MLP combined with DG methods provides quite desirable performances in controlling numerical oscillations as well as capturing key flow features.

1 Introduction Multi-dimensional limiting process (MLP) has been developed quite successfully in finite volume methods (FVM). Compared with traditional limiting strategies, such as TVD or ENO-type schemes, MLP effectively controls unwanted oscillations particularly in multiple dimensions. The theoretical foundation of the MLP limiting strategy is to satisfy the maximum principle to ensure multi-dimensional monotonicity. A series of researches [2, 4, 6] clearly demonstrates that the MLP limiting strategy possesses superior characteristics in terms of accuracy, robustness and efficiency in inviscid and viscous computations on structured and unstructured grids in FVM. Recently, discontinuous Galerkin (DG) methods become more popular as a higher-order discretization of hyperbolic conservation laws because of its own merits, such as flexibility to handle complex geometry, compact stencil for higher-order reconstruction, and amenability to parallelization and hp-refinement. However, one of the major bottlenecks in DG methods is to design a robust, accurate and efficient limiting strategy to handle oscillations and discontinuities in multiple C. Kim (B) School of Mechanical and Aerospace Engineering, Institute of Advanced Aerospace Technology, Seoul National University, Seoul 151-744, Korea e-mail: [email protected]


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dimensions. Although there have been several remarkable efforts to control oscillations using TVB-based limiters or WENO-type limiters, overall performances are not satisfactory at all in terms of accuracy and/or efficiency, especially in controlling oscillations near shock discontinuities in multi-dimensional flows. Based on the remarkable successes of the MLP in FVM, the MLP limiting philosophy is now extended into the DG framework to provide an accurate, efficient and robust limiting strategy.

2 Multi-dimensional Limiting Process 2.1 Multi-dimensional Limiting Condition In order to maintain multi-dimensional monotonicity, the present limiting strategy exploits the MLP condition, which is an extension of the one-dimensional monotonic condition. The basic idea of the MLP condition is to control the distribution of both cell-centered and cell-vertex physical properties to mimic a multi-dimensional nature of flow physics. We focus on the observation that well-controlled vertex values at interpolation stage make it possible to produce monotonic distribution of cell-averaged values. This observation is verified by showing that cell-centered and cell-vertex values reconstructed by the MLP limiting safisfy the maximum principle. Based on the observation, the vertex values are required to satisfy the following MLP condition min max ≤ qvt x ≤ q¯neighbor , q¯neighbor

(1)

min max are the minimum and max, q¯neighbor where qvt x is the vertex value, and q¯neighbor imum cell-averaged values sharing the same vertex point. The MLP condition can be implemented regardless of grid topology, though the present work is focused on triangular mesh. For example, the detailed implementation in FVM can be seen in the work of Ref. [2, 6] for structured grids and Ref. [4] for unstructured grids. The effectiveness of the MLP condition is supported by the maximum principle, which plays a key role in ensuring the monotonicity in multiple dimensions. Compared to previous approaches, the MLP condition fully exploits the cell-averaged values sharing the same vertex point as well as edges, so the MLP limiting is less sensitive to local mesh distribution and accurately represents multi-dimensional flow physics.

3 Extension of MLP into Discontinuous Galerkin Methods In DG methods, the distribution within a cell is approximated by the sum of shape modes in a suitable function space

Multi-dimensional Limiting Process for Discontinuous Galerkin Methods

q hj (x, t) =

n

(i)

q j (t)b(i) (x),

181

(2)

i=1

where q hj is an approximated solution of q(x, t) on the cell T j and b(i) is a shape function. In the present work, the RKDG method with orthogonal shape functions is adopted. In order to prevent unwanted oscillations near discontinuities, limiting procedure is essential. Especially for efficient and accurate computations, limiter should be selectively activated on the troubled-cells only. Thus, an accurate troubled-cell marker, as well as a limiter, is crucial to obtain an accurate monotone solution in the DG framework. In FVM, the MLP condition is used to identify and control the maximumprinciple-violating cells [4]. If property distribution is linear, the edge midpoint (or any quadrature point) can be restricted by controlling the vertex points where extrema occur. However, for higher-order reconstruction greater than P1, it is not guaranteed. The P1-based MLP condition may not identify the troubled-cells which may lead to violation of the maximum principle. After some analysis, a more strict condition is found to be necessary to identify the troubled cells. The MLP-based troubled-cell marker is proposed using the following augmented MLP condition ≤ qvh,min ≤ qvhi , qvhi ≤ qvh,max ≤ q¯vmax , q¯vmin i i i i

(3)

, q¯vmax are minimum where qvhi is an approximated solution at vertex vi and q¯vmin i i and maximum among the cells sharing the vertex. If distribution at vertex violates the above condition, this cell is marked as a troubled cell. To avoid the clipping problem across local smooth extrema, a simple but effective extrema detector is additionally introduced as follows Δq¯vi = q¯vmax − q¯vmin ≤ KΔx 2 , i i

(4)

where K is a parameter to be determined. Numerical experiments strongly indicate that computed results are not sensitive to the change of K , and its optimal value is around 100. With the augmented MLP condition (Eq. (3)) and the extrema detector (Eq. (4)), the troubled-cells are marked and the MLP slope limiter is applied only on these cells with the following distribution q˜ hj (x, t) = q¯ j + φ M L P ∇q j · (x − xj ).

(5)

For the Euler systems, density or entropy variable is used in the troubled-cell marker to identify physical discontinuities. After marking the troubled-cells, the MLP limiting is applied on conservative variables, the same as in the MLP in FVM.

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4 Numerical Result 4.1 Compressible Flow with Sinusoidal Density Perturbation In order to examine accuracy in continuous flow, the Euler system with the following smooth initial data is considered (ρ0 , u 0 , ν0 , p0 ) = (1 + 0.2 sin (π(x + y)), 0.7, 0.3, 1.0).

(6)

The computational domain is [0, 2] × [0, 2] with periodic boundary condition. Triangular elements are created by dividing uniform square elements along the diagonal. Table 1 shows the result of grid refinement test at t = 2. The MLP limiting combined with the MLP troubled-cell marker (K = 100) preserves the designed accuracy in DG reconstruction. Table 1 Grid refinement test for compressible flow with sinusoidal density perturbation DG-P1, MLP-u1, K=100 DG-P2, MLP-u1, K=100 Grid

L∞

Order

L1

Order

L∞

Order

L1

Order

10 × 10 × 2 20 × 20 × 2 40 × 40 × 2 80 × 80 × 2

1.774E-2 3.411E-3 7.019E-4 1.543E-4

– 2.38 2.28 2.18

1.046E-2 1.919E-3 3.743E-4 7.965E-5

– 2.45 2.36 2.23

8.137E-4 1.046E-4 1.326E-5 1.664E-6

– 2.96 2.98 2.99

4.279E-4 5.242E-5 6.527E-6 8.159E-7

– 3.03 3.01 3.00

4.2 A Mach 3 Wind Tunnel with a Step This is one of the popular cases to test higher-order high-resolution schemes. Around the expansion corner, computational meshes are slightly clustered without any special treatment. Lax-Friedrich scheme is applied as a numerical flux. Figure 1 shows the density contours computed on triangular grids of h = 1/160 at t = 4.0. DG reconstructions with the MLP limiter provide monotonic solutions with a sharp capturing of the slip line from the shock triple point.

P1, MLP-u2, K = 5, h = 1/160

P2, MLP-u2, K = 5, h = 1/160

Fig. 1 Comparison of density contours for the Mach 3 wind tunnel with a step. Thirty equally spaced contour lines from ρ = 0.32 to ρ = 6.15

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4.3 Strong Vortex–Strong Shock Interaction Shock-vortex interaction generally leads to complex but challenging flow patterns. When the rapidly rotating vortex strikes the strong shock, vortex core is severely elongated and eventually split by compression [5]. The computational domain is [0, 2] × [0, 1] and the normal shock wave with Ms = 1.5 is located at x = 0.5. A composite vortex is located at (x c , yc ) = (0.25, 0.5), and the velocity profile is given as follows ⎧ r ⎪ ⎨ vm a a vθ = vm a 2 −b 2 r − ⎪ ⎩ 0

b2 r

if r ≤ a if a ≤ r ≤ b , if r > b

(7)

where (a, b) = (0.075, 0.175), and the maximum Mach number of angular velocity is 0.9. AUSMPW+ scheme [1] is used as a numerical flux. In Fig. 2, numerical Schlieren images computed by FVM and DG reconstructions with the MLP limiter are compared at t = 0.7 on coarse and fine grids. While FVM-MLP on coarse grid does not show the vortex-splitting phenomenon, DG P2-MLP captures this flow structure very clearly. In addition, DG reconstruction with the MLP limiter provides a more detailed resolution for emitted sound waves and discontinuities.

FVM, MLP-u1, h = 1/100

DG-P1, MLP-u1, K = 10, h = 1/100

DG-P2, MLP-u1, K = 10, h = 1/100

FVM, MLP-u1, h = 1/400

DG-P1, MLP-u1, K = 10, h = 1/400

DG-P2, MLP-u1, K = 10, h = 1/400

Fig. 2 Comparison of numerical Schlieren images of strong vortex–strong shock problem at t = 0.7

4.4 Interaction of Shock Wave with 2-D Wedge The computational domain contains a regular triangle with length L = 1 on [−2.5, 4.6] × [−2.5, 2.5]. The tip of wedge is located at the origin. As an initial condition, the moving shock with Ms = 1.34 is located at x = 0. RoeM flux scheme [3] is applied. Figure 3 shows the comparison of numerical Schlieren images at t = 3.25. Computed results confirm again that the DG reconstruction with the MLP limiter guarantees a sufficient resolution to capture complex shock-vortex structure.

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FVM, MLP-u1

DG-P1, MLP-u1

DG-P2, MLP-u1

Fig. 3 Comparison of numerical Schlieren images on interaction of shock wave with 2-D wedge at t = 3.25 (Bottom left corner of each image: Close-up view around the primary vortex)

5 Conclusion Guided by the MLP condition and the maximum principle [4], the Multidimensional limiting process is efficiently and accurately designed within discontinuous Galerkin framework. The proposed approach is able to accurately capture complex multi-dimensional flow structure without yielding unwanted oscillations. Various numerical results show the desirable characteristics of the proposed limiting strategy, such as multi-dimensional monotonicity, improved accuracy and efficiency. Acknowledgements Authors appreciate the financial supports provided by NSL (National Space Lab.) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (Grant 20090091724), and by National Institute for Mathematical Sciences.

References 1. Kim, K.H., Kim, C., Rho,O.H.: Methods for the accurate computations of hypersonic flows, part I: AUSMPW+ scheme. J. Comput. Phys. 174, 38–80 (2001) 2. Kim, K.H., Kim, C.: Accurate, efficient and monotonic numerical methods for multidimensional compressible flows: Part II: Multi-dimensional limiting process. J. Comput. Phys. 208, 570–615 (2005) 3. Kim, S., Kim, C., Rho, O.H., Hong S.K.: Cures for the shock instability: Development of a shock-stable Roe scheme, J. Comput. Phys. 185, 342–374 (2003) 4. Park, J.S., Yoon, S-.H., Kim, C.: Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids. J. Comput. Phys. 229, 788–812 (2010) 5. Rault, A., Chiavassa, G., Donat, R.: Shock-vortex interactions at high Mach numbers. J. Sci. Comput. 19, 347–371 (2003) 6. Yoon, S-.H., Kim, C., Kim, K.H.: Multi-dimensional limiting process for three-dimensional flow physics analyses. J. Comput. Phys. 227, 6001–6043 (2008)

Runge Kutta Discontinuous Galerkin to Solve Reactive Flows Germain Billet and Juliette Ryan

Abstract A Runge–Kutta Discontinuous Galerkin method (RKDG) to solve the reactive Navier–Stokes equations written in conservation form and with no restrictive physical hypothesis is presented. Real thermodynamics laws are taken into account. A particular care has been taken to solve correctly the stiff gaseous interfaces. Some 1-D and 2-D test cases are presented.

1 Introduction Phenomena which develop in the combustion chambers or industrial furnaces are multiple. Injections of fuel and oxidizer, premixed or not, in a strongly turbulent flow set up gas pockets which react generating either premixed flames or diffusion flames together with intense acoustic phenomena. These physical processes have to be captured in order to describe in details the reactive flows. It is then necessary to take into account the transport coefficients for the parabolic operator of the Navier–Stokes (NS) equations and detailed chemical kinetics for the production rates (source operator). Vortex dynamics, transport, stretching and wrinkling of fronts as well as acoustic propagation must be suitably resolved by the hyperbolic operator. Within the framework of the Euler equations, we found two recent articles [8, 11] which propose to solve the two-medium flow simulations starting from a RungeKutta Discontinuous Galerkin (RKDG) method. Both use the level set method to compute the location of the interface. The advantage is to keep a conservative treatment of the interface but the difficulty lies in the coupling between a traditional RKDG method far from the interface and a Discontinuous Galerkin (DG) level set method around the interface. The level set method is very powerful and is used in many fields but its main weakness is that it can only treat nondiffusive interfaces. Its use in the field of the reactive flows is thus limited to the follow-up of thin front with very reductive assumptions as to the reactive processes. For the reacting porous G. Billet (B) ONERA, BP72 FR-92322, Chatillon Cedex, France e-mail: [email protected]


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media, some approaches using DG to solve reacting NS equations have been developed but generally the incompressibility hypothesis is introduced, see for example [12]. In addition, many papers have presented the resolution of NS equations with RKDG method for nonreactive flows with a constant specific heat ratio γ . But when the fluid contains Ns gaseous species in a flow where the temperature T evolves strongly as in reactive flows, it is necessary to take into account the variation of γ and to assume γ = γ (Yi , T) where Yi is the mass fraction of the ith species (i ∈ S = [1, Ns]). In the same way, the transport coefficients depend on the same variables. We describe here a RKDG method with no restrictive physical hypothesis to solve the reactive Navier–Stokes equations written in conservation form. For the time integration, a third-order TVD Runge-Kutta scheme is used [10]. One test case is presented with DGP1 and DGP2 which simulates 2-D premixed flames H2 -air with a simplified kinetic scheme.

2 The Discontinuous Galerkin Approach This approach is based on the work of Cockburn and Shu (see [4]). For simplicity, we shall just recall the 1D case, representative of all dimensions. The solution as well as the.test function space is given by / Whk = ϕ ∈ L ∞ (Ω) / ∀ j, ϕ|Ω j ∈ P k (Ω j ) where P k (Ω j ) is the space of polynomials of degree ≤ k on the * cell Ω j = [x j−1/2 , x+j+1/2 ] = Δx j . We define a local (l) orthogonal basis over Ω j , φ (l) j (x), l = 0, 1, ..., k where φ j (x) are the Legendre

polynomials. The numerical solution in the test function space Whk is written as ∀t ∈ [0, T ], ∀x ∈ Ω j , U h (x, t) =

l=k

(l)

(l)

U j (t)φ j (x) for x ∈ Ω j

l=0

where U j(l) (t) are the degrees of freedom. A weak formulation of the problem is obtained by multiplying the N S equations by a test function ϕ and by integrating on each cell Ω j . Then, a discrete analogous is obtained by replacing the exact solution U by the approximation U h (x, t), ∀t ∈ [0, T ], ∀ j, Ωj

∂U h (x, t) ϕ(x) dx + ∂t

Ωj

∂F (U h (x, t)) ϕ(x) dx = ∂x

Ωj

ϕ(x)S (U h (x, t))d x.

The test function ϕ is replaced by each element of the basis set φ(l) j (x) and the inviscid and viscous fluxes are integrated by part. The source term S represents the production rates of each species.

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3 Hyperbolic Operator The main difficulty is to capture correctly the gaseous interfaces. If no care is taken to solve the hyperbolic part of the Navier–Stokes equations in conservative form, spurious pressure oscillations develop when several species diffuse through an interface where γ varies [1, 6]. Some models were proposed in the past but, either they present other disadvantages like the appearance of oscillations on other quantities, or they are written for particular cases of front capturing or for mixtures where CP is supposed constant. In 2001, Abgrall and Karni [2] proposed the DF model for mixtures where CP depends only on the mass fractions. In 2003, Billet and Abgrall [3] generalized this approach for CP = CP (Yi , T ). This approach, which is no longer strictly conservative but only quasi-conservative for the hyperbolic part of NS equations when γ varies, suppresses the numerical oscillations of the physical quantities through the gas interfaces. DF is efficient only with numerical fluxes which can be extended to a complex thermodynamic laws γ = γ (Yi , T ) as for example HLLC [13]. A limiting treatment based on Krivodonova’s paper [5] is applied for DGP2 when the reactive processes are activated.

4 Parabolic Operator The recovery method reproduces the symmetric DG formulation with natural penalty terms depending on the accuracy of the method. Van Leer’s idea is to construct for each piecewise continuous polynomial basis of degree k defined on two adjacent cells a unique continuous polynomial space of degree 2k +1 on the union of the two cells. In consequence, to the approximation of the solution as an expansion in the discontinuous basis functions locally corresponds an identical expansion in the smooth recovery basis. Thanks to this new smooth basis, the diffusion fluxes across the cell interfaces can be naturally computed. The details of this approach can be found in [15] for the 1-D problems and the extension to 2-D is developed in [14]. In reactive flows, the transport coefficients are not uniform and consequently the diffusive part of the NS equations are integrated by part once only.

5 Numerical Example 5.1 Draughtboard Reactive Mixing H2 -Air in a Shear Flow We are interested in a draughtboard mixture of H2 -Air (simulating the region of an idealized multi-point injector). This mixture is immersed in a sinusoidal temperature field varying between 300 K and 1,200 K. This gaseous mixing is also submitted to in a sinusoidal velocity field varying for each component u and v between −100 ms−1 and +100 ms−1 . With these values, the Mach number M varies in the

188


range −0.4 < M < 0.4 during the computation. Four shear lines are present in the flow at t = 0 (x = 0 m, x = 0.05 m, y = 0.025 m and y = 0.075 m). The domain dimensions are L 2 = 0.1 m × 0.1 m and the boundary conditions are periodic. A Cartesian grid is used with Δx = Δy. The mixture wave length L λY = L/4 and the wave length of temperature and velocity fields are L λT = L uλ = L vλ = L. At t = 0, T T T = 750 − 450 λ sin 2πu y/L λ sin 2πu x/L u = −100 sin 2π x/L λ cos 2π y/L λ v = −100 sin 2π x/Lvλ cos 2π y/Lvλ Y O2 = 0.117 1 + sin 2π x/L λY sin 2π y/L λY Y H2 = 0.015 1 − sin 2π x/L λY sin 2π y/L λY Y N2 = 1 − Y O2 − Y H2 . Some initial fields are plotted in Fig. 1. The time step δt = 5×10−8 s corresponds to Cfl = 0.1. The kinetic scheme is made up of 4 species and 2 Arrhenius reactions [9]: H2 + O2 ↔ 2OH and H2 + 2OH ↔ 2H2 O. 1 Figures 2 and 3 present the time evolution of the temperature with DGP and DGP2 on a same grid L λY = 8Δx . At t = 10−3 s, the solutions are similar but more details appear with DGP2 . But at t = 2 × 10−3 s, the topology of the temperature field is quite different and the temperature levels are no longer the same. These differences are due to the very strong acoustic waves that develop at the beginning of the reactions. DGP2 , thanks to its weaker dissipation, captures better these strong fluctuations and gives after t = 10−3 s a different shape of the flames in the domain.

(a)

(b)

Fig. 1 Initial values. (a) Initial hydrogen mass fraction, (b) initial velocity and temperature fields (K )

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(a)

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(b)

Fig. 2 Temperature fields with DGP . (a) Temperature field at t = t = 2 × 10−3 s 1

(a)

10−3 s,

(b) temperature field at

(b)

Fig. 3 Temperature fields with DGP . (a) Temperature field at t = t = 2 × 10−3 s 2

10−3 s,

(b) temperature field at

6 Conclusion A RKDG approach with no restrictive physical hypothesis has been developed for reactive flows. This approach is stable and robust and works well for low Mach number and subsonic flows. Its extension to supersonic combustion is under development. Comparisons with a sixth order finite difference DNS code with detailed kinetic scheme [7] are to be carried out.

References 1. Abgrall, R.: How to prevent pressure oscillations in multicomponent flows: A quasi conservative approach. J. Comput. Phys. 125, 150–156 (1996) 2. Abgrall, R., Karni, S.: Compressible multifluids. J. Comput. Phys. 169, 594–623 (2001)

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3. Billet, G., Abgrall, R.: An adaptive shock-capturing algorithm for solving unsteady reactive flows. Comput. Fluids. 32, 1473–1495 (2003) 4. Cockburn, B., Shu, C.-W.: The Runge-Kutta Discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141, 199–224 (1998) 5. Krivodonova, L.: Limiters for high-order discontinuous Galerkin methods. J. Comp. Phys. 226, 879–896 (2007) 6. Larouturou, B.: How to preserve the mass fraction positive when computing compressible multicomponent flows. J. Comput. Phys. 95, 59–84 (1991) 7. Laverdant, A., Thevenin, D.: Interaction of a gaussian acoustic wave with a turbulent premixed flame. Comb. Flame. 134, 11–19 (2003) 8. Qiu, J., Liu, T., Khoo, B.C.: Runge-Kutta Discontinuous Galerkinmethods for compressible two-medium flows simulations: One-dimensional case. J. Comput. Phys. 222, 353–373 (2007) 9. Rogers, R.C., Chinitz, W.: Using a global hydrogen-air combustion model in turbulent reacting flow calculations. AIAA J. 21(4), 586–592 (1983) 10. Shu, C.-W., Osher, S.: Efficient implementation of, essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 73–85 (1998) 11. Sollie, W.E.H., van der Vegt, J.J.W., Bokhove, O.: A space-time discontinuous galerkin finite element method for two-fluid problems. Memorandum 1849, University of Twente, Enschede (2007) 12. Sun, S., Wheeler, M.F.: Analysis of discontinuous Galerkin for multicomponent reactive transport problems. Comp. Math. Appl. 52, 637–650 (2006) 13. Toro, E.F., Spruce, M., Spears, W.: Restoration of the contact surface in the HLL Riemann solver. Shock Waves. 4, 25–34 (1994) 14. van Leer, B., Lo, M., van Raalte, M.: A discontinuous Galerkin method for diffusion based on recovery. 18th AIAA CFD conference (2007) 15. van Raalte, M., van Leer, B.: Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Proceedings ICFD 2007 (2007)

Comparison of the High-Order Compact Difference and Discontinuous Galerkin Methods in Computations of the Incompressible Flow Artur Tyliszczak, Maciej Marek, and Andrzej Boguslawski

Abstract High-order compact difference scheme (CD) based on the half-staggered mesh is compared with discontinuous Galerkin method in computations of the incompressible flow. Assessment of the accuracy is performed based on the classical test cases: Taylor–Green vortices, Burggraf flow and also for temporally evolving shear layer. The CD method method provides very accurate results with expected order of accuracy, 4th and 6th. Similarly for the discontinuous Galerkin method provided that the number of degrees of fredom is close to the number of nodes in computations with CD method. Furthermore, it appeared that CD method is much more efficient than the discontinuous Galerkin method of comparable accuracy.

1 Introduction The most accurate spectral or pseudospectral methods [1] based on Fourier or Chebyshev approximation allow for detailed study of complex fundamental physical phenomena. Their application is however limited to simple geometries and meshes defined by the collocation points which means that in geometrically complicated problems they are no longer feasible. The coordinate transformation combined with the domain decomposition are not always possible but even if so, this approach considerably complicates the numerical codes and sometimes leads to additional problems related to stability, singularity, etc.. The compact difference schemes [6] can be regarded as an alternative of the spectral approach sharing their most important properties (accuracy, resolving efficiency). Additionally, they can be easier applied in the cases of non-uniform and non-cartesian grids with various type of the boundary conditions. The finite volume, the finite element or discontinuous Galerkin approach are examples of the methods without any particular geometrical or mesh related limitations. In this work we compare two of the above mentioned methods, the compact difference method and the discontinuous Galerkin method applied to the incompressible flow calculations. A. Tyliszczak (B) Czestochowa University of Technology, 42-200 Czestochowa, Poland e-mail: [email protected]


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2 Numerical Methods Both for the discontinuous Galerkin method and for the compact difference method we apply the projection method [3] which after the pressure correction step enforces the divergence free velocity field. In the compact difference (CD) approach we use half-staggered meshes [5, 10] where the pressure is shifted half cell size with respect to the velocity field. This approach eliminates the pressure oscillation caused by the pressure-velocity decoupling occurring on the collocated meshes. In the halfstaggered approach additional computational cost and complications are related to the mid-point interpolation and mid-point derivative approximation, however we note that these steps are less computationally expensive than in the case of the fully staggered approach. We also note that shifting the pressure half-cell size from the velocity node is relatively easy and can be surely applied to any code based on the collocated meshes – that was done in our case. Another important point concerns solution of the Poisson equation (introduced by the projection method) which in the case of the high-order methods can be very expensive computationally as the resulting coefficient matrix of the Poisson equation is dense. Solving the Poisson equation we combine a low-order pressure gradient discretization with an explicit high-order discretization of the divergence operator. The resulting discretization has similar resolving characteristics as the compact scheme but it leads to 5 or 7-diagonal system which can be effectively solved by a direct method for sparse systems (analogy of TDMA algorithm). The discontinuous Galerkin method (DGM) belongs to the family of finite element methods (FEM) and allows for employment of unstructured flexible computational meshes. Moreover, hp-adaptivity is much easier to implement, comparing to classical continuous Galerkin method, as the method supports non-conforming elements (hanging nodes) and possibility of various expansion bases (shape functions). The locality of DGM makes it an ideal framework for parallelisation. The matrices, that are typically constructed in FEM (e.g. mass matrix), can be calculated seperately for each element and their size is related to the number of local degrees of freedom. DGM has also some drawbacks, the main being larger number of variables comparing to continuous Galerkin method and additionally the second order derivatives (e.g. viscosity, diffusion terms, Laplacian) have to be handled by mixed methods. In the present work the implementation of DGM method has been developed for 2D incompressible, viscous flow. The code accepts unstructured FEM meshes with quadrilateral elements. Such meshes offer better quality, comparing to those with triangular elements. Also the basis functions may be constructed in a straightforward way by tensor products of one-dimensional functions [2, 4, 8]. The disadvantage is that the Jacobian of transformation from the standard element to the physical element is not constant (in general), which results in considerable storage requirements – DGM matrices must be constructed for each of the elements seperately. The second order terms, representing viscosity and Laplacian of the pressure, are handled by LDG (Local Discontinuous Galerkin) method. The solver of Poisson equation for the pressure is based on simple, iterative method.

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3 Results The comparison of the accuracy and efficiency of the compact and discontinuous Galerkin methods is performed for classical test cases, i.e. Taylor–Green vortices, Burggraf flow and also for temporal shear layer flow. Both codes were written in the Fortran 90 with mathematical libraries (Lapack/BLAS) used wherever possible. The test of the efficiency showed that the code based on CD scheme is considerably faster than the code based on DG method. For instance, for the CD scheme, the computations of the Taylor–Green flow (simulation time equal to 1.0 with Δt = 10−5 ) on the mesh 100 × 100 took about 45 min. For the DG method the accuracy of the results similar to that obtained with CD was achieved with 20 × 20 elements with 4th order of expansion, in this case the computations took 10 h, approximately.

3.1 Taylor–Green and Burggraf Flows The Taylor–Green or Burggraf flow are examples of the test cases for which the analytical solution of the Navier-Stokes equations exists and therefore these cases are used in this work to assess the order of the applied methods. Figure 1 shows parts of the computational domains with contours of the horizontal velocity component obtained with 6th order compact difference method with 3th and 4th order approximation for the near boundary and boundary nodes. Figure 1 on the right hand side presents the error decreasing with the number of mesh points N . The error is defined as: 0 1 Ny 1 Nx 2 1 1 1 u c (xi , y j ) − u a (xi , y j ) Err or = 2 Nx N y

(1)

i=1 i=1

Taylor-Green vortices

u(x)

Burggrafflow

0

10

–1

10

10-2 ~ 1/N6 Error

10-3 10-4 -5

10 0.040 0.030 0.020 0.010 0.000 -0.010 -0.020 -0.030 -0.040

0.900 0.700 0.500 0.300 0.100 0.004 -0.000 -0.016 -0.100 -0.200 -0.260

~ 1/N4

-6

10

10-7 10-8 40 N

80

120 160

Fig. 1 Horizontal velocity component in the Taylor–Green flow and in the Burggraf flow (left figure) – CD results. Error of the horizontal velocity component in Taylor–Green flow (line with triangles) and in the Burggraf flow (line with squares)

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10

–1

10

DGM (P = 1) DGM (P = 2)

–2

10

–2

10

2nd ord.

–4

10

Error

Error

2nd ord.

–3

10

–3

10

–4

10

3rd ord.

–5

10

4th ord.

–6

–5

10

3rd ord.

10

–7

10

–6

10

10–7

DGM (P = 1) DGM (P = 2) DGM (P = 3) DGM (P = 4)

10–1

5th ord.

10–8 –9

10 10

20

40 60 N (CD), Ndof(DGM)

80

110 140

(a)

10

20

40 60 N (CD), Ndof(DGM)

80

110 140

(b)

Fig. 2 Error of the horizontal velocity component for the Burggraf flow (left figure) and Taylor– Green flow. Number of degrees of freedom Ndof = Nelx (P + 1), Nelx – number of elements in one direction

where u c and u a are computed and analytical value respectively. We note that in the computations performed with CD scheme for the velocity components we applied Dirichlet boundary conditions resulting from the analytical solution – typically (and also in our implementation of the discontinous Galerkin method) the Taylor flow is solved with the periodic boundaries which eliminate the influence of the lower order boundary closure scheme. Besides that, as the number of the boundary nodes is small compared to all computational nodes one may observe that for the Taylor flow the error decreases according to the assumed order of the scheme. Influence of the boundary closure is seen in the case of the Burggraf flow for which the solution accuracy decreases to 4th order. The results obtained with discontinous Galerkin method indicate that for Burggraf flow the method is only second order accurate, even when the order of expansion P is larger than one. The error decay is presented in Fig. 2. The reduction of order of accuracy for this particular flow is known in the literature and probably is related to the specific type of projection method used in the implementation of DGM. Reduction of the order has not been observed in CD code, using somewhat different procedure in projection step. In the case of Taylor vortices, the order of accuracy agrees with expectations, i.e. it equals P + 1. It should be noticed, that for high order expansion DGM offers better accuracy than CD of the same number of degrees of freedom.

3.2 Temporal Shear Layer Flow The temporal shear layer flow is relatively simple example of turbulent flow, it was often used as the test case for the numerical codes for Direct and Large Eddy Simulation [7, 9, 11, 12]. The velocity profile is defined as u(y) = Uerf (yπ 1/2 ). The Reynolds number defined as: Re = Uδ/ν (U – free stream velocity, δ – initial width of shear layer, ν – viscosity) was equal 250. In the computations with CD method

Comparison of the High-Order Compact Difference and DGM t=0

195

t = 22

t = 40

Fig. 3 Evolution of shear layer (vorticity isocontours)

the mesh consisted of 128 × 128 uniformly distributed nodes. The evolution of the shear layer (vorticity isocontours) is presented in Fig. 3, details of the flow characteristics may be found in [7, 9]. In this chapter we concentrate on the comparison of the amplitudes of the most unstable mode and its subharmonic. The length of the computational domain L was equal four times the length of the most unstable mode (equal to 2.32π [9]), allowing to form four vortices corresponding to the most unstable mode which then was pairing due to the presence of the subharmonic modes. The amplitudes of a given mode of the initial perturbation or in the evolved field are measured by the integrated RMS of the velocity modes defined as: 3 Aα =

+L/2

−L/2

41 2uˆ x (α)uˆ ∗x (α)dy

2

(2)

where uˆ i (α) – Fourier coefficient of velocity component u x and uˆ ∗x (α) – its complex conjugate. The evolution of the most unstable mode A1 and its subharmonic A1/2 are shown in Fig. 4. It can be observed the agreement between DGM and CD results is satisfactory only when the number of degrees of freedom in DGM is at least the same as the number of gridpoints in CD, although for the subharmonic mode reasonable accuracy is obtained even for quite coarse mesh (20 × 20, P = 2). This is not surprising, as the characteristic spatial scales related to that mode are much 0.7

2 1.8

A1 (DGM 80 x 80, P = 1) A1 (DGM 20 x 20, P = 2) A1 (CD)

0.6 0.5

1.6 1.4 1.2

A1

A1/2

0.4 0.3 0.2 0.1 0

1 0.8 0.6

A1/2 (DGM 80 x 80, P = 1)

0.4

A1/2 (DGM 20 x 20, P = 2) A1/2 (CD)

0.2 0

20

40

60 time

80

100

0 0

20

40

60

80

time

Fig. 4 Evolution of the most unstable mode (left figure) and its subharmonic (right figure)

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larger than for the most unstable mode. Both codes provide correct moment of time in which the maximum of the mode A1 is attained [7, 9]. Acknowledgements The support for the research was provided within the research grant WZ1-101-701/2008 founded by Polish Ministry of Science and the statutory funds BS-1-103301/2004/P. The authors are grateful to the Cyfronet and TASK Computational Center for access to the computing resources.

References 1. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. Springer-Verlag, Berlin Heidelberg (1988) 2. Cockburn, B., Shu, C.-W.: Runge-Kutta Discontinuous Galerkin Methods for convectiondominated problems, J. Sci. Comput. 16(3) (2001) 3. Fletcher, C.A.J.: Computational Techniques for Fluid Dynamics. Springer-Verlag, Berlin Heidelberg (1991) 4. Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press, Oxford (2005) 5. Laizet, S., Lamballais, E.: High-order compact schemes for incompressible flows: A simple and efficient method with quasi-spectral accuracy. J. Comput. Phys. 228, 5989–6015 (2009) 6. Lele, S.K.: Compact finite difference with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992) 7. Lesieur, M., Staquet, C., LeRoy, P., Compte, P.: The mixing layer and its coherence examined from the point of view of two-dimensionla turbulence. J. Fluid Mech. 192, 511–534 (1988) 8. Li, B.Q.: Discontinuous Finite Elements in Fluid Dynamics and Heat Transfer. Springer, London (2006) 9. Moser, D.R., Rogers, M.M.: The three-dimensional evolution of plane mixing layer: pairing and transition to turbulence. J. Fluid Mech. 247, 275–320 (1993) 10. Nagarajan, S., Lele, S.K., Ferziger, J.H: A robust high-order compact method for large eddy simulation. J. Comput. Phys. 19, 392–419 (2003) 11. Tyliszczak, A.: Influence of the compact explicit filtering method on the perturbation growth in temporal shear-layer flow. J. Theoretical Appl. Mech. 41, 19–32 (2003) 12. Vreman, B., Geurts, B., Kuerten, H.: On the formulation of the dynamic mixed subgrid-scale model. Phys. Fluids. 6, 40–57 (1994)

On the Boundary Treatment for the Compressible Navier–Stokes Equations Using High-Order Discontinuous Galerkin Methods Andreas Richter and Jörg Stiller

Abstract An appropriate boundary treatment is one of the most important tasks to perform when carrying out numerical simulations. The technique to define the boundary condition depends strongly on both the numerical scheme and the type of differential equation to be solved. In terms of implementation effort and cost of computational resources, every boundary should be treated locally in both space and time. In this chapter we discuss techniques to deal with adiabatic walls in the framework of high-order discontinuous Galerkin methods for compressible flow.

1 Introduction The treatment of boundaries is essential for the numerical simulation of fluid mechanics problems. A multitude of requirements exist: (i) Stability and robustness. (ii) The boundary condition has to reflect the physical problem, e. g. Dirichlét conditions and Neumann conditions. (iii) The formulation should be local in space and time. (iv) The boundary treatment has to respect the numerical scheme, e. g. the representation of curved walls has to be of the same order as the underlying scheme. (v) The implementation has to fulfill the requirements listed above without an unrealistic expansion of the computational domain. Because there are so many requirements, it is difficult to find a general solution. For the investigation of aeroacoustic problems as well as musical woodwind instruments we use a high-order discontinuous Galerkin finite element method in conjunction with a TVD Runge-Kutta time integration scheme to solve the unsteady and compressible Navier-Stokes equations in the form = ∇ · D(, ∇), ∂t U + ∇ · F(U)

A. Richter (B) Institute for Aerospace Engineering, Technische Universität Dresden, 01062 Dresden, Germany e-mail: [email protected]


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with U = (ρ, ρ v, ρet )T the solution vector and = ( v , T )T the vector of primitive variables. To solve the diffusive terms an interior penalty scheme is used [1, 4–7]. This leads to the weak form

Ωe

Γe

+ −

1 2

Γe

e ) − D( e , ∇e ) dΩe ve ∂t Ue dΩe − ∇ve · F(U ve Hc U± + Hd {e } , {∇e } , ± dΓe e ,n e ,n n U− − − + dΓe = 0 ∇ve · C e e e

(1)

(for ∀ ve ∈ Ve and ∀ Ωe )

D(, ∇) = C() · ∇ * + c + i p {Cnn } (+ − − ) Hd = n · D Δ with ve the test function, Hc the convective and Hd the diffusive fluxes at the element boundaries Γe . The averages and jumps are defined as follows: {q} =

1 − 1 − q } = ( (q + q + ) ; { q + q + ) 2 2

[[q]] = n(q − − q + ) ; [[q]] = n · ( q − − q + ) , using {}− to denote values coming from the element interior and {}+ for values from the outside. The convective and diffusive fluxes and the derivatives of the primitive variables need an appropriate treatment. Different strategies exist: 1. Hard correction of the solution field. This approach strictly keeps the boundary value but leads to instabilities, because it does not respect the type of the underlying differential equation. 2. Direct definition. This is only possible if the exact flux is known, e. g. at walls. 3. Alternatively, the outer values at the corresponding edge can be defined in such a way that the resulting numerical flux meets the exact flux. 4. Characteristic treatment. Following Hirsch [2] or Polifke et al. [3], the solution can be divided into variables that propagate along characteristic directions. This quasi one-dimensional approach respects the hyperbolic character of the differential equation, but may fail at more complex geometries. 5. Puffer zone techniques such as Perfectly Matched Layers (PML) extend the computational domain by additional zones, in which the solution is damped to a homogeneous mean flow. Depending on the boundary type, one of the strategies can be a good choice. For convective and diffusive fluxes, a combination of the listed strategies or a different strategy altogether may be useful or necessary. In this contribution we focus on adiabatic walls. We shall discuss inflows and outflows in further publications.

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2 Treatment of Adiabatic, Full-Reflective Walls 2.1 Advective Flux Treatment In terms of adiabatic and hard walls, the characteristic formulation reads Z 1new − Z 1old 1 1 ! = Δt Z 1 = Δt p corr − Δt vnbc = Δt Z 4 = Δt p + Δt vn , Δt ρa ρa ˙ It describes the reflection of characteristics running outwith Δt Z = ΔtR−1 U. wards. This formulation requires time derivatives of both the pressure and velocity, which are typically determined by linearization. Because this linearization can produce oscillations that disturb the solution, this strategy shall not be pursued further. At stationary walls the boundary flux can be prescribed directly with ⎛

0

⎞

Hbc = ⎝ p − n⎠ , 0 or alternatively it can be constructed by mirroring the outer state to fit the boundary flux ⎞ ⎛ ρ− U+ = ⎝ v − α1 nvn ⎠ α1 = 1 . . . 2, β1 = −1 . . . 1 . (2) − ρet− + β1 ekin For frictionless systems – like solutions of the Eulerian equation – the outer total energy state fully accounts for the inner energy. For systems with friction, different choices for the kinetic energy are possible.

2.2 Diffusive Flux Treatment The characteristic formula to treat the diffuse flux is also based on time linearization and is therefor not discussed here. As the velocity gradients are unknown at the wall, only an incomplete data set to define the exact diffusive flux is given. Alternatively the outer state can be constructed by setting the velocity field to zero or by mirroring the velocity. The outer temperature has to meet the inner temperature because no temperature gradients are allowed. The constructions scheme follows + =

− v − α2 v− , T−

with α2 equal to one in terms of setting the velocity to the boundary value, or two for mirroring.

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2.3 Gradients Gradients can be treated similarly as diffusive fluxes. Only the temperature gradient is given to construct the outer state, while the wall shear stress is unknown. This leads to (∇ v)+ = (∇ v)− ∂n T + = ∂n T − − β3 ∂n T − , with β3 also a constant value between one and two.

3 Test Case As test case, a lid-driven cavity is investigated in two steps: First the flow is driven by a moving isothermal wall on the top. Then, all walls are defined stationary and adiabatic and the development of the mass and the total energy inside the container and also the temperature gradients at the boundaries are measured. The domain is divided into 4 × 4 elements of polynomial order 8, the Reynolds number during the start up process is 400 (Fig. 1). Fig. 1 Lid-driven cavity. Snapshot of flow field

4 Mass Conservation Table 1 displays mass conservation as function of advective flux treatment. Two results emerge: A factor β1 = 0 in Eq. (2) significantly reduces the conservation properties. They also deteriorate when the outer normal velocity is set to zero instead of mirroring it. On the other hand, neither the treatment of diffusive fluxes nor that of velocity and temperature gradients influences mass conservation measurably.

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Table 1 Mass conservation as function of adiabatic fluxes∗ Advective flux treatment L ∞ norm of mass error H = Hbc vn+ = −vn− vn+ = −vn− vn+ = −vn− vn+ = 0 vn+ = 0 vn+ = 0 ∗ Diffusive

ρet+ ρet+ ρet+ ρet+ ρet+ ρet+

= = = = = =

ρet− ρet− ρet− ρet− ρet− ρet−

− + ekin − − ekin − + ekin − − ekin

7.320 · 10−12 8.301 · 10−9 8.286 · 10−9 7.320 · 10−12 5.606 · 10−7 4.099 · 10−7 4.852 · 10−7

fluxes equivalent to the first row in Table 2.

Table 2 Total energy conservation as function of primitive variables and their gradients∗ . Diffusive flux L ∞ norm of total treatment Gradients energy error v+ v+ v+ v+

=0 =0 = − v− = − v−

∗ Advective

T+ − T− T+ − T− T+ − T− T+ − T−

∂n T + ∂n T + ∂n T + ∂n T +

=0 = −∂n T − =0 = −∂n T −

(∇v)+ (∇v)+ (∇v)+ (∇v)+

= (∇v)− = (∇v)− = (∇v)− = (∇v)−

1.628 · 10−6 4.443 · 10−6 9.818 · 10−6 7.844 · 10−8

flux H = Hbc .

5 Total Energy Conservation The other hand the total energy strongly depends on the treatment of the diffusive terms. For reasons of mass conservation the energy correction term ±ekin drops and the remaining choices of the advective flux have no effects on the total energy. Interestingly, uncoupled variations of both the diffusive flux and the primitive variable gradients seem to have no effect on the energy losses. Only one configuration – the combination of mirrored temperature gradients and mirrored velocities – pushes the energy losses two orders downward.

6 Conclusion In this article we present results of the study of different techniques to define adiabatic hard walls in the framework of high-order discontinuous Galerkin schemes. In these, mass conservation depends strongly on the exact definition of the advective flux, while energy conservation is guaranteed only if both the primitive variables and their gradients are properly defined. In general, mirroring all values may be a good choice for fluxes that cannot be calculated directly. Acknowledgements This work was supported by the German National Science Foundation (Deutsche Forschungsgemeinschaft, DFG) within the project Numerical simulation of the sound spectrum and the sound radiation in and around a recorder (Numerische Simulation des Klangspektrums und der Schallausbreitung in und um eine Blockflöte).

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References 1. Hartmann, R., Houston, P.: An optimal order interior penalty discontinuous Galerkin discretization of the compressible Navier-Stokes equations. J. Comput. Phy. 227(22), 9670–9685 (2008) 2. Hirsch, C.: Numerical Computation of Internal and External Flows, vol. 1 & 2. WileyInterscience Publication, New York, USA (1990) 3. Polifke, W., Wall, C., Moin, P.: Partially reflecting and non-reflecting boundary conditions for simulation of compressible viscous flow. 213, 437–449 (2006) 4. Richter, A., Brußies, E., Stiller, J., Grundmann, R.: Aeroacoustic investigation of woodwind instruments based on discontinuous Galerkin methods. In: Computational Fluid Dynamics Review 2010. World Scientific Publishing Company, Inc., 5 Toh Tuck Link, Singapore (2010) 5. Richter, A., Stiller, J., Grundmann, R.: Stabilized discontinuous Galerkin methods for flowsound interaction. J. Comput. Acoustics 15(1), 123–143 (2007) 6. Richter, A., Stiller, J., Grundmann, R.: Stabilized high-order discontinuous Galerkin methods for aeroacoustic investigations. ICCFD 2008, Seoul (2008) 7. Stiller, J., Richter, A., Brußies, E.: A physically motivated discontinuous galerkin method for the compressible navier-stokes equations. ICOSAHOM 2007, Beijing (2007)

Analytical and Numerical Investigation of the Influence of Artificial Viscosity in Discontinuous Galerkin Methods on an Adjoint-Based Error Estimator Jochen Schütz, Georg May, and Sebastian Noelle

Abstract Recently, it has been observed that the standard approximation to the dual solution in a scalar finite difference context can actually fail if the underlying forward solution is not smooth (Giles and Ulbrich, Convergence of linearised and adjoint approximations for discontinuous solutions of conservation laws. Technical Report, TU Darmstadt and oxford university, oxford, 2008). To circumvent this, it has been proposed to over-refine shock structures of the primal solution. We give evidence that this is also the case in the discontinuous Galerkin approach for the one-dimensional Euler equations if one explicitly adds diffusion. Despite this, on the first sight very negative result, we demonstrate that, if using the dual solution only for adaptation purposes, a special treatment seems not to be necessary to get good convergence in terms of a target functional.

1 Introduction 1.1 Background Distributing degrees of freedom economically in the numerical computation of hyperbolic conservation laws has motivated the development of error control. Traditionally, e.g. in the context of elliptic equations, one has always tried to reduce the solution error in some given norm below some given treshold, i.e. the approach take as little degrees of freedom as possible to achieve w − wh < ε, where w is the exact and wh the approximated solution (see e.g. [10] for a good overview). In engineering applications, however, one is often interested in only one or a few single numbers coming from the solution w, e.g. in aerodynamics these numbers could be lift or drag. Mathematically speaking, one is interested in J (w), where J. Schütz (B) AICES Institute for Advanced Study in Computation Engineering Science, RWTH Aachen University, 52062 Aachen, Germany e-mail: [email protected]


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J is a given, probably nonlinear functional (usually called target functional). This has motivated the aproach take as little degrees of freedom as possible to achieve |J (w) − J (wh )| < ε. This error can be approximately calculated via the adjoint method. A lot of work has been put into the development and theoretical justification of adjoint methods supposed that the underlying solution is smooth ([4, 8]). Despite its importance for the use in the context of hyperbolic conservation laws, only a few publications have been concerned with the correct adjoint formulation in the case where there is a jump discontinuity in the underlying forward solution, see e.g. [3, 9, 13] and the references therein. Recently, it has been observed that the discrete approximation to the dual can actually fail [7, 9] in the case of a shock in the underlying forward solution. The key observation of the authors of the aforementioned paper is that the loss of information within the discontinuity is too much to allow for giving precise initial data of the gradient of the objective function. To circumvent this feature, the authors in [9] have proposed (and, for their very special setting, also proved) that an over-refinement of the shock-structure, i.e. giving viscosity of O(h α ), where h is the cell-size and α < 1, does lead to convergent adjoint solutions. We have been conducting experiments and came to the conclusion that this does also hold for the case when one approximates the one-dimensional steady-state Euler equation with a DG method – stabilized by explicit artificial diffusion – so, as expected, it is not a particular feature of the approach of [7] and [9]. We will demonstrate these findings and investigate how they influence an adaptation algorithm.

1.2 Underlying Primal Equation We will in the course of this chapter consider a one-dimensional model problem; the quasi one-dimensional steady Euler nozzle flow with flux f and sourceterm S ([1]) through a convergent-divergent duct of given geometry A(x) (x ∈ Ω ⊂ R).

1.3 Adaptive DG Discretization with Artificial Diffusion We discretize our primal equations by using a Discontinuous Galerkin method (cf. [2, 5]). In the last few years, these methods have gained quite a lot of attention, as they bring together the advantages of Finite Element methods (the builtin high-order) and the advantages of Finite Volume methods (the stability due to upwinding). Furthermore, even in the very high order context, DG methods stay local which makes them very well suited for parallel computations. Recently, several authors (e.g. [6, 12]) have suggested to explicitly add artificial, solution-dependent diffusion terms for stabilization into the method. We discretize the convective term with the approach of Cockburn and Shu [5], while for the viscous term, we use a Bassi – Rebay 2 discretization [2].

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We modify the treatment of the boundary conditions in such a way that our discretization is asymptotically adjoint consistent (cf. [2, 11]). In our adaptation procedure, we start with a rather coarse grid and calculate a primal solution wh1 . We then calculate a dual error estimator (see next section) which drives the adaptation algorithm. To avoid “extremal” refinement steps (like e.g. refine only one cell, or refine all), we use the so-called fixed-fraction criterium ([4]) which refines a fixed fraction q of cells that have the largest error indicator.

2 Linearization and Dual Equation We are mainly interested in variations of J , i.e. (directional) gradients of the func5 tional J (w) := Ω p(w) d x. We will for simplicity in this section only deal with the one-dimensional model problem (for more sophisticated cases, we refer to [3] and the references therein), where its solution is assumed to have a discontinuity at x = α. Given a function v that is supposed to fulfill the dual equation − f (w)T vx + S (w)T v = p (w)

v f (w)ξ = 0 T

∀x ∈ Ω

(1)

∀x ∈ ∂Ω, ∀ξ ∈ K er (B (w)),

A(α) v2 (α) = − A (α)

(2) (3)

and that is additionally supposed to be continuous at x = α, we end up with the fact that the linearization can be written as J (w)w =

Ω

v T · ( f (w)w)x + S (w)w d x.

(4)

Of course the internal boundary condition (3) is awkward in actual numerical calculations for several reasons, one being the uncertainty about α. This is why most authors do not at all consider this internal boundary condition but just calculate a solution to (1)–(2) in the hope of reaching (3) for free. This has been very successful in the context of low-order methods (cf. [8, 14]), although Giles published, for the time-dependent scalar case, a simple counter example (cf. [7]).

3 Numerical Results The h α viscosity approach: As mentioned in the introduction, the convergence towards the dual solution can actually fail unless one gives enough diffusion ([7, 9]). We have – in the 1D Euler case – been conducting experiments and have found out that this is also true in our setting.

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Let us consider a one-dimensional test-case with a free-stream Mach number of 0.5 and an exit pressure of pout = 1.0. For our smooth geometry, this creates a shock at position x = 5.25... (which can, thanks to the explicit solution, be calculated in advance). Both primal and dual solution are calculated with polynomial order once p = 1 and once p = 3 on the same mesh. To demonstrate our findings, we assume that the viscosity ε(w) ≡ ε is constant throughout our domain Ω. As a measure of error, we use the relative deviation of the dual solution v in fulfilling (3). Figure 1 plots the amount of viscosity used versus the error in the dual solution. There are some remarks in order about the plot: Notice the right hand side where all the graphs lie above each other. This means that both primal and dual solution are perfectly grid-converged and that the only source of error stems from the viscosity

Fig. 1 Amount of artificial viscosity versus error in the adjoint solution for p = 1 and p = 3


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term. The interesting point is always when the graphs “leave” the common region (on the left hand side of the plot) because this is where discretization errors begin to dominate the overall error. What can be seen is that these occurences do not appear linearly with respect to the mesh-size but that they eventually occur earlier and earlier. These findings motivate the conclusion that one shoud take viscosity constant to h α , where α is slightly less than unity, to ultimately guarantee convergence of the adjoint solution. We demonstrate the advantage in taking the viscosity proportional to h 0.8 in Fig. 2. When giving viscosity proportional to h, the dual solution does not converge

Fig. 2 Plot of the model problem and convergence history of the adjoint solution for p = 3

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at all, while when adding diffusion as O(h 0.8 ), we have a perfectly acceptable convergence history. What one can also see when comparing the two pictures in Fig. 1 is that the higher-order solution needs a little less (about half) the amount of viscosity to accurately resolve the dual solution on the same mesh. It remains to prove these findings similar to [9]. Unfortunately, this will be - due to the more complicated structur of the Euler equations – a non-trivial task as already mentioned in [9]. Adaptivity by the adjoint without enforcing the h α approach: Despite the very negative results in the preceeding subchapter, we want to demonstrate that – for using adaptivity – it does not seem necessary at all to enforce the dual boundary condition. Let us therefore look at Fig. 3. We ran a test with the same starting parameters as the one for Fig. 2, with the difference that we now did not refine uniform, but we did refine where the adjoint error estimation did tell us to. The constant viscosity was chosen to be of size O(h) and O(h 0.8 ), respectively, where h is the minimum mesh-size. Of course we are not primarily interested in whether the adjoint internal boundary condition (3) does converge or whether not, but we are interested in how far the primal functional J converges. It can be seen clearly from Fig. 3, that even if the dual does not converge towards the boundary condition (and, in so far, the dual does not converge properly),we still get an adaptation criterion that seems to be very reasonable. Note that, furthermore, due to the viscosity of size O(h), the “standard” approach which concerns viscosity seems to converge mush faster in terms of the functional

Fig. 3 Convergence history of the adjoint solution for p = 3


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J (which is – after all – not very surprising). The actual surprising part is that as an adaptation criterion, the standard approach seems to be as good as the overrefinement approach. This in some way demonstrates the capability of the adjoint approach even if one does not enforce or even achieve the internal dual boundary condition (3). (This is what actually most authors do – just neglect (3). Our findings could justify their approach.) Acknowledgements Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged.

References 1. Anderson, J. D.: Fundamentals of Aerodynamics. McGraw-Hill, New York, NY (2001) 2. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.L.: Unified analysis of discontinuous galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2002) 3. Bardos, C. and Pironneau, O.: Data assimilation for conservation laws. Methods Applications Anal. 12, 103–134 (2005) 4. Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica. 10, 1–102 (2001) 5. Cockburn, B., Shu, C.-W.: The Runge-Kutta Local Projection P 1 -discontinuous Galerkin Finite Element Method for Scalar Conservation Laws. RAIRO Model. Math. Anal. Number. 25, 337–361 (1991) 6. Barter, G., Darmofal, D.L.: Shock capturing with higher-order, pde-based artificial viscosity. Proc. of the 18th AIAA CFD Conference, Miami, FL, AIAA-2007-3823 (2007) 7. Giles, M.B.: Discrete adjoint approximations with shocks. In Conference on Hyperbolic Problems, Springer-Verlag (2002) 8. Giles, M., Pierce, N.: An introduction to the adjoint approach in design. Flow, Turbulence and Combustion, 65, 393–415 (2000) 9. Giles, M., Ulbrich, S.: Convergence of linearised and adjoint approximations for discontinuous solutions of conservation laws. Technical Report, TU Darmstadt and Oxford University, Oxford (2008) 10. Oden, J.T., Ainsworth, M.: A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142, 1–88 (1997) 11. Oliver, T.A., Darmofal, D.L.: Analysis of dual consistency for discontinuous galerkin discretizations of source terms. SIAM J. Numer. Anal. (2008) 12. Persson, P.-O., Peraire, J.: Sub-cell shock capturing for discontinuous galerkin methods. Proc. of the 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA-2006-112 (2006) 13. Ulbrich, S.: Optimal control of nonlinear hyperbolic conservation laws with source terms. Technical Report, TU München (2001) 14. Venditti, D.A., Darmofal, D.L.: Adjoint error estimation and grid adaptation for functional outputs: Application to quasi-one-dimensional flow. J. Comput. Phys. 164, 204–227 (2000)

Part VIII

Vortex Dynamics

Analysis of a Swept Wind Turbine Blade Using a Hybrid Navier–Stokes/Vortex-Panel Model Kensuke Suzuki, Sven Schmitz, and Jean-Jacques Chattot

Abstract A 10% backward sweep is added to the NREL wind turbine blade and its aerodynamic performance is studied using a hybrid Navier–Stokes/Vortex-Panel solver. A modified twist distribution for the swept blade obtained with an optimization vortex code is tested and a higher power gain is achieved in the low wind speed case of 7 ms−1 compared to the NREL Phase VI rotor. However the backward sweep increased the bending moment. For more relevant comparison with the aerodynamic performance of the straight blade, adjustment of the blade setting angle is done to achieve the same bending moment. It is observed that the backward sweep will improve the stability of the blade for gust control while keeping other aerodynamic performances at or above that of the NREL rotor.

1 Introduction Responding to an increased interest in advanced blade tip design of wind turbine blades to get more power at lower speed and to get better gust control, a backward (BW) sweep is added to the National Renewable Energy Laboratory (NREL) Phase VI rotor. In an earlier work, Chattot [2, 3], wind turbine blades with optimal chord and twist distributions and equipped with the S809 airfoil are analyzed and the aerodynamic performances are compared to the straight blades, using a vortex line method (VLM). In this chapter, we extend the analysis to a hybrid method that confines a Navier-Stokes (NS) solver to a small region around a wind turbine blade and represents the far-field by the aforementioned vortex method. In the past, the hybrid solver has shown fast, robust and accurate power predictions for the NREL Phase VI rotor under steady and zero yaw conditions, see Schmitz and Chattot [5–7]. The computational domain size of the NS solver is only a few chord lengths around the rotor blade. Rigid helicoidal vortex sheets are attached to the trailing edges of the blades and extend to the Trefftz plane, which is typically located ten K. Suzuki (B) University of California, Davis, CA, USA e-mail: [email protected]


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Fig. 1 (a) NS grid (left); (b) lifting line and helicoidal vortex sheets (middle); (c) 10% BW swept NREL blade (right)

blade radii downstream of the rotor. Figure 1a shows the grid of the NS zone, where the wake surface is aligned with the first 60◦ of azimuth of the helicoidal vortex sheet in Fig. 1b. The spanwise distribution of bound circulation inside the NS zone serves as input for the vortex model, which in turn provides induced velocities at the outer boundary of the NS zone using the classical Biot–Savart law. A converged solution is obtained in about six coupling steps. For the current study, CFX V12.0 and FLUENT V12.0.16 have been coupled with the vortex method and compared to the NREL Phase VI Rotor configuration experiments.

2 Method 2.1 Coupling Process and Convergence for NREL Blade The initial solution is calculated applying uniform flow velocity to the outer boundary of the NS zone. A rigid helicoidal vortex sheet modeled with the power output is generated. In this work, the spanwise circulation is calculated by integration of pressure on airfoil surface using the pressure difference on the upper and lower surfaces, Eq. (1). The outer boundary velocity of the first coupling is set as the sum of the induced velocity, incoming wind velocity and the blade rotation entrainment velocity. The coupling process is repeated until the circulation distribution converged to a tolerance. Figure 2a shows convergence history of the spanwise circulation. Table 1 is a comparison between force and power from integration of surface pressure and solver summary output. It is seen that the torque from the pressure integration is consistent with the value from the solver summary. Figure 2b indicates that the circulation is consistent with that determined by the Kutta–Zhukovsky lift theorem. Figure 3 shows the capture of the blade tip vortex inside the NS solution and on the exit surface of the domain boundary calculated by the induced velocity.

Γ (x/c) =

Ulocal {C p + − C p − } 2

where Ulocal is the local velocity at the blade section.

(1)

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Fig. 2 (a) Spanwise circulation history of coupling process (left); (b) spanwise circulation from Eq. (1) and from normal force (right)

NREL blade Torque [J] Power [W] % of NREL ak

− ω;

b Spalar t

Table 1 NREL blade torque/power CFX (Fluent) Pressure CFXa (Fluentb ) Pressure + integration Pressure moment viscous moment –816.4 –818.6 –763.2 (–886.8) (–885.6) (–840.5) 6155.7 6172.2 5754.3 (6686.7) (6677.4) (6337.0) +1.4 % +1.6 % –5.2 % (+10.1 %) (+10.0 %) (+4.4 %)

− Allmaras.

Fig. 3 Capture of tip vortex inside NS solution and on the exit boundary

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3 Results and Discussion The earlier VLM results on optimization indicate that the BW sweep gives a favorable bias compared to forward sweep. The optimization was performed for both twist and chord distributions while keeping constant thrust on the tower. The optimum twist distribution is also tested but with the chord distribution of NREL blade.

3.1 Backward Swept NREL Blade The 10% BW sweep is applied to the NREL blade with a shearing transformation, Fig. 1c. The spanwise circulation was not significantly changed from NREL blade, as seen in Fig. 5a and the efficiency gain in Table 2 is less than 1%. However, the pitching moment increased 283%. The bending moment increased 0.8%. The positive pitching moment means “nose-down” and twists the blade towards lower incidences for gust control, as indicated in the earlier work with VLM.

3.2 Backward Swept Blade with Optimum Twist Distribution The optimized twist and chord distributions are given by VLM in Fig. 4. For this study, the twist distribution of the NREL blade is curve fitted with 9th order polynomial. The circulation increased near the blade tip, in Fig. 5b and the efficiency improved 2.64% (see Fig. 6 for pressure distribution). The pitching moment increased 293%. The bending moment and thrust increased 4.3, 2.8% respectively.

Table 2 Result summary

Swept blade Blade setting Angle [deg.] Thrust [N] Bending moment [J] Pitching moment [J] Torque [J] Power [W] Efficiency Efficiency % increase from NREL blade a Same

CFX swept opt-twistb

CFX NREL (fluent)

3 1290 –

3 3 1202.1 (1295.2) 1204.3 −4143.2 (−4439.0) −4176.4

3 1236.2 −4321.4

3.084 (ref. 3.094) 1194.9 −4143.2

3.468 (ref. 3.3) 1186.2 −4143.2

–

60.7 (64.4)

171.5

177.8

170.4

171.5

−805.3 6072.3 0.364 –

−763.3 (−840.5) 5755.2 (6337.0) 0.345 (0.379) –

−767.3 5785.7 0.346 0.53

−783.3 5907.0 0.354 2.64

−762.7 5751.6 0.344 −0.06

−760.4 5732.8 0.343 −0.39

blade setting angle; setting angle

b Adjusted

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Fig. 4 Optimum chord (left) and twist (right) distributions obtained with VLM

0

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Fig. 5 (a) Spanwise circulation: swept NREL blade vs. NREL blade (left); (b) spanwise circulation: swept blade with modified twist vs. swept NREL blade (right)

−Cp r/R=80% 3 10% BW sweep of NREL blade, Upper 10% BW sweep of NREL blade, Lower 10% BW sweep with optimum twist, Upper 10% BW sweep with optimum twist, Lower Straight NREL blade, experiment

2.5 2

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Fig. 6 Pressure coefficient distribution at 80% blade section: swept NREL blade (square, circle) vs. swept blade with modified twist (diamond, triangle) and straight NREL blade experiment (star)

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3.3 Adjustment of the Blade Setting Angle The adjustment of the blade setting angle is introduced to keep the same bending moment as the straight blade. Forces are estimated by the linear interpolation from results at nearby setting angles. Table 2 shows pitch moment improved about 280% with keeping overall aerodynamic performance. As commented above, the twist distribution is associated with the optimum chord in Fig. 4, keeping the thrust on the tower. The optimum chord distribution would reduce the thrust and must be taken into account in the future work.

4 Conclusions The addition of sweep to the NREL blade does not change significantly the power or bending moment but is expected to improve the blade static stability with respect to wind gusts. The optimum twist distribution contributes to a small increase in power capture at low speeds. Acknowledgements The support of General Electric for this research is gratefully acknowledged.

References 1. Munk, M. M.: The Minimum Induced Drag of Aerofoils. NACA Rept. No. 121 (1921) 2. Chattot, J.J.: Effects of Blade Tip Modifications on Wind Turbine Performance Using Vortex Model. Computers Fluids. 38, 1405–1410 (2008) 3. Chattot J.-J.: Optimization of wind turbines using helicoidal vortex model. J. Solar Energy Eng. 125(4), 418–424 (2003) 4. Chattot, J.J.: Design and analysis of wind turbines using helicoidal vortex model. J. Comput. Fluid Dyna. 11(1), 50–54 (2002) 5. Schmitz, S., Chattot, J.J.: A parallelized coupled Navier-Stokes/Vortex-Panel Solver. ASME J. Solar Energy Eng. 127, 475–487 (2005) 6. Schmitz, S., Chattot, J.J.: A coupled Navier-Stokes/Vortex-panel solver for the numerical analysis of wind turbines. Computers Fluids. 35, 742–745 (2006) 7. Schmitz, S., Chattot, J.J.: Characterization of three-dimensional effects for the rotating and parked NREL phase VI wind turbine. ASME J. Solar Energy Eng. 128, 445–454 (2006)

Using Feature Detection and Richardson Extrapolation to Guide Adaptive Mesh Refinement for Vortex-Dominated Flows Sean J. Kamkar, Antony Jameson, Andrew M. Wissink, and Venkateswaran Sankaran

Abstract The article describes a Cartesian-based adaptive mesh refinement approach applied to vortex-dominated flows. Several distinct feature-detection methods are investigated to furnish a means for tagging cells for refinement. In each case, appropriate normalization is defined so that the process is automated for a range of operating conditions. Richardson extrapolation is proposed to assess the local error and terminate the mesh refinement once adequate error reduction is achieved.

1 Introduction Many aerodynamics flowfields require the accurate resolution of vortices over long distances. Adaptive Mesh Refinement (AMR) provides an attractive means of achieving the desired accuracy in an efficient manner. Because the vortex transport occurs predominantly in the so-called “off-body” region of the flowfield, we adopt an overset-based dual-mesh paradigm [8] that utilizes unstructured grids in the near-body region and Cartesian meshes in the off-body region. The unstructured meshes provide ease of grid generation for complex geometries and accurately capture boundary layer effects. The Cartesian meshes allow the use of high-order discretization and ease of automation of the AMR process. The present article is focused on the development and implementation of algorithms for Cartesian-based AMR. The approach is based upon using feature-detection methods [6] for tagging off-body regions for refinement and Richardson extrapolation techniques [1, 7] for estimating the local error and terminating the refinement process. Specific attention is placed on automating the AMR process so that the methods can be applied to a variety of practical flowfields without need for user intervention. S.J. Kamkar (B) Stanford University, Palo Alto, CA 94305, USA e-mail: [email protected]


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2 Feature Detection Four feature detection methods are presented and tested in this work: (1) the Q-criterion [4], (2) the λ2 criterion [5], (3) the eigenvalues of the velocity gradient tensor ∇u [2], and (4) the correlation between the symmetric and antisymmetric parts of ∇u [3]. All methods are presented in a non-dimensional form by imposing a local normalization, which emphasizes generality and automation. Specifically, the goal is to avoid any problem-dependent parameters that need to be tuned. For each method, a function, f val is defined, which is used for purposes of vortex detection. A given cell is marked for refinement when the local value of the function exceeds a pre-specified threshold value, i.e., f val > tval . Non-dimensional Q: Using both the rotational (Ω) and strain (S) components of the velocity gradient tensor, i.e., S = (∇u + ∇u)/2, Ω = (∇u − ∇u)/2, we can obtain a measure of the relative vortical strength [5]: fval =

1 2

||Ω||2 − 1 ||S||2

(1)

Non-dimensional λ2 : It can be shown that the tensor, S 2 + Ω 2 , approximately represents the pressure field in a flow. Therefore, the eigenvalues of this system can be used to determine pressure minima, which usually occur in or near vortex cores. If the eigenvalues are ordered in a fashion where λ1 ≥ λ2 ≥ λ3 , the condition λ2 < 0 identifies the vortex core [6]: f val = −

λ2 ||S||2

(2)

Modified-: The eigenvalues of ∇u reveal information about the local flow-field. For example, if a pair of complex eigenvalues is detected, swirling motion is present [2]. Furthermore, the magnitude of the complex conjugate (λci ) is a measure of the vortical strength: λci ||S||

(3)

λ+ − 1. ||S||2

(4)

f val =

S-Ω Correlation: The three previous methods have been designed to pinpoint vortex cores; here, we examine a method that attempts to locate a vortex sheet. Specifically, we leverage the fact that strain and rotation rates are correlated, and that both have large magnitudes in vortex sheets [3]. λ+ is the largest eigenvector orthogonal to the vorticity vector for the system SΩ − Ω S: f val =

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In all the above methods, the functional value is used to mark mesh cells for refinement if its value is greater than the pre-specified threshold value, i.e., f val > tval . Each threshold function has a zero offset so that, in all cases, positive values represent regions of swirl. As an example, we consider the application of the modified- method applied to the AMR-tracking of a propagating Lamb vortex. Figure 1 shows a snapshot of the solution at a particular time instance. The vortex is observed to be completely enclosed within the fine-grid region. Furthermore, as the solution advances through the domain, the fine grid properly tracks the feature and provides adequate resolution so that it is well resolved. At the end of the simulation, even after the vortex has travelled about 12 chord lengths, its initial coherent structure and strength remains relatively intact. Figure 1 also compares the non-dimensional Q, λ2 , , and S-Ω methods with the traditional vorticity-tagging method (tval = 8.67 × 10−3 ). A solution computed on a uniformly fine mesh is also included. Additionally, the exact solution is provided for further validation. It is clear that the different feature detection techniques agree very closely with the exact solution as well as with the numerical predictions from the uniform fine mesh. It is important to note that, while the vorticity method also performs well, the vorticity magnitude has to be tuned for optimal performance for each case, while in the normalized methods proposed here no user-adjustments are necessary. Runtime performance is catalogued in Table 1. The adaptive methods are observed to be about ten times faster than the comparable uniform-fine grid

0.09 Uniform Fine

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0.085 0.08 0.075

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Fig. 1 Modified- method applied to advecting vortex case (left). Comparison of methods after a single period (right) Table 1 Comparison of average runtime per time-step and number of grid points for the single advecting vortex case on uniform and adaptive grids Coarse uniform Medium uniform Fine uniform Non-dim Q (adaptive)

n

Sec/time-step

1.53 × 104

0.0742 0.534 4.19 0.305

1.15 × 105 8.93 × 105 1.04 × 105

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computation. The coarse and medium uniform grid solutions, on the other hand, are observed to be highly dissipative in comparison.

3 Richardson Extrapolation-Based Error Estimation The local truncation error (L T E) of the numerical scheme can be used as an indicator of the local error. L T E can be computed by using the difference between the exact and discrete solutions, i.e., u n+1 − wn+1 O(x p t k ). = O(x p ) + O(t q ) + t q−1

(5)

k=1

This is verified in Fig. 2, where 5th-order accurate spatial discretization is observed for the advecting Lamb vortex case when it is run for several different values for t and x. Note that, for a given t, spatial refinement shows an initial fifth order error decrement, but eventually the temporal errors dominate and the error is observed to level off. Smaller t runs are observed to lead to a third-order accurate reduction in the temporal errors. A major drawback of using the L T E to estimate the local error is that one typically needs an exact solution to develop an error estimate for the discrete solution. In practical cases where exact solutions do not exist, it is conceivable that the exact solution may be substituted by a discrete equivalent computed on a highly refined mesh. This concept describes the fundamental idea behind the Richardson extrapolation error estimation process.

L2 Norm of x−Momentum (ρ u) LTE

10–2

10−4 dt = 5 dt = 5/2

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dt = 5/4 dt = 5/8 dt = 5/16

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slope = 5.0

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Fig. 2 L T E spatial convergence behavior of x-momentum (ρu) for the advecting vortex case

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We can define the Richardson error estimation as follows: E h/2 =

wh/2 − wh + O(h p+1 ). 2p − 1

(6)

Note that the denominator is a function of p, which is the order of accuracy employed by the finite difference scheme and is constant for the different grid levels. Indeed, the error expression is simply the difference between the solutions on the coarse and fine grids. As in the case of the L T E, a similar accuracy analysis can be performed for the Richardson error estimator. It can be shown that, regardless of the number of refinement levels which exist between a coarse and fine solution, the error expression of Eq. (6) is − wcn+1 w n+1 f t

p

≈ −λxc +

λ2 t p xc , 2

(7)

10

−2

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−4

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−6

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dt = 5 dt = 5/2 dt = 5/4 dt = 5/8 dt = 5/16 dt = 5/32 dt = 5/64 dt = 5/128 dt = 5/256 dt = 5/512

slope = 5.0

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10 dx

0

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L2 Norm of x−Momentum (ρ u) (Finest vl)

which indeed exhibits pth-order spatial accuracy. It is interesting to note the similarity to the unsteady L T E of Eq. (5). However, the Richardson error estimation contains only the spatial errors since the temporal errors are the same for the two grids and therefore cancel out. Thus, as the mesh is continually refined, the Richardson error estimator continues to converge at the rate corresponding to the spatial scheme. This behavior is illustrated by Fig. 3 where a coarse solution is compared against an extremely fine mesh. Furthermore, Fig. 3 also illustrates the case where the fine solution is calculated from the solution of the next level of refinement (rather than the highest level of refinement). This result also shows fifth order accuracy, demonstrating that the next level solution can be reliably used to define the local error. Additional testing of the above AMR approach for rotorcraft applications will be addressed in future work. 10

−2

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dt = 5 dt = 5/2 dt = 5/4 dt = 5/8 dt = 5/16 dt = 5/32 dt = 5/64 dt = 5/128 dt = 5/256 dt = 5/512

slope = 5.0

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0

10 dx

Fig. 3 Richardson spatial convergence behavior of x-momentum (ρu) for the advecting vortex case using a fine solution that has six levels of refinement. The plot on the left uses the finest level solution for calculating the error. The plot on the right uses the next level solution for calculating the error

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4 Conclusion This chapter has evaluated four distinct feature detection methods designed to drive Cartesian-based AMR in the off-body domain for vortex-dominated flows. Using traditional dimensional techniques to identify local regions for refinement, such as vorticity magnitude or the Q-criterion, can be problematic due to the requirement of a user-defined threshold which controls the refinement. Four particular non-dimensional schemes are evaluated: Q, λ2 , Modified-, and S-Ω correlation. It is shown that all the non-dimensional methods match the desirable performance without need for any parametric tuning. Additionally, the adaptive methods produce solutions that are comparable to the uniform-fine mesh equivalent, but are an order of magnitude faster. This chapter also lays the groundwork for using a Richardson Extrapolation error estimator to further control refinement for time-dependent flows. It is proposed that this error be utilized to provide a termination criterion to the AMR process. Future work will apply this feature-detection-based AMR methodology to a broader range of vortex-dominated flow-fields, particularly rotorcraft aeromechanics computations. Acknowledgements Material presented in this chapter is a product of the CREATE-AV Element of the Computational Research and Engineering for Acquisition Tools and Environments (CREATE) Program sponsored by the U.S. Department of Defense HPC Modernization Program Office. The article has been approved for public release by the AMRDEC Public Affairs Office (FN4345).

References 1. Aftosmis, M.J., Berger, M.J.: Multilevel error estimation and adaptive h-refinement for Cartesian Meshes with embedded boundaries. 40th AIAA Aerosciences Conference, Reno, AIAA Paper 2002-0863 (2002) 2. Chong, M.S., Perry, A.E., Cantwell, B.J.: A general classification of three-dimensional flow fields. Phys. Fluids 2, 765–777 (1990) 3. Horiuti, K., Takagi, Y.: Identification method for Vortex sheet structures in turbulent flows. Phys. Fluids 17(12), (2005) 4. Hunt, J.C.R., Wray, A.A., Moin, P.: Eddies, streams, and convergence zones in turbulent flows. In its Studying Turbulence using Numerical Simulation Databases, 2. Proceedings of the 1988 Summer Program, pp. 193–208, Stanford (SEE N89-24538 18-34) (1988) 5. Jeong, J., Hussain, F.: On the identification of a Vortex. J. Fluid Mech. 285, 69–94 (1995) 6. Kamkar, S.J., Jameson, A.J., Wissink, A.M., Sankaran, V.: Feature-driven Cartesian adaptive mesh refinement in the Helios code. 48th AIAA Aerosciences Conference, Orlando, AIAA Paper 2010-0171 (2010) 7. Nemec, M., Aftosmis, M.J., Wintzer, M.: Adjoint-based adaptive mesh refinement for complex geometries. 46th AIAA Aerosciences Conference, Reno, AIAA Paper 2008-725 (2008) 8. Wissink, A.M., Sitaraman, J., Sankaran, V., Pulliam, T., Mavriplis, D.: A multi-code pythonbased infrastructure for overset CFD with adaptive Cartesian grids. 46th AIAA Aerosciences Conference, Reno, AIAA Paper 2008-927 (2008)

Triple Decomposition Method for Vortex Identification in Two-Dimensional and Three-Dimensional Flows V. Koláˇr, P. Moses, and J. Šístek

Abstract A new vortex-identification method has been recently proposed, named as Triple Decomposition Method (TDM). By decomposing the local motion of a fluid into straining, shearing and rigid-body rotation, it particularly allows to eliminate the biasing effect of shear. The paper presents an application of TDM to numerical flow data, both in two and three dimensions.

1 Introduction A new method for the identification of a vortex was proposed by Koláˇr in [5] and applied in [6]. It has been named Triple Decomposition Method (TDM) due to the fact that the motion of a fluid is locally considered a superposition of three elementary motions – straining, shearing and rigid-body rotation – each of them corresponding to a certain part of the velocity gradient tensor ∇u. Thus, proper decomposition of ∇u enables separation of the vorticity associated with rigid-body rotation from the vorticity caused by shear. Exploiting this fact in vortex identification, TDM eliminates the shear component arguing that the rotational part of the residual represents a vortex more conveniently since it is a direct kinematic measure of the swirling motion, whereas the results obtained by standard double decomposition technique may be distorted by the shear-induced vorticity.

2 Triple Decomposition Method The velocity gradient tensor considered in terms of the triple decomposition can be expressed as a sum of three quantities corresponding to basic types of motion, ∇u = S R E S + R E S + ∇u S H ,

(1)

V. Koláˇr (B) Institute of Hydrodynamics, Academy of Sciences of the Czech Republic, CZ-16612 Prague 6, Czech Republic e-mail: [email protected]


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where S R E S represents the residual strain-rate tensor and R E S the residual vorticity tensor, both “corrected” by the extraction of shear into a separate term ∇u S H . Recall that in double decomposition only strain-rate tensor S = 12 (∇u+(∇u)T ) and vorticity tensor = 12 (∇u − (∇u)T ) are used, both containing the effect of shear. The residual vorticity tensor R E S is the quantity proposed by TDM as the proper kinematic measure of a vortex. The sum S R E S + R E S forms a so-called residual tensor, obtained from ∇u as ⎞ ⎛ (sgn u y )MIN(|u y |, |vx |) . ux residual vy . ⎠ , = ⎝(sgn vx )MIN(|u y |, |vx |) tensor . . wz

(2)

using simplified notation u x = ∂u ∂ x etc.; the non-specified off-diagonal elements are constructed analogously. The effective shear, i.e. the maximum impact of shear extraction, occurs at the minimum of the norm of the residual tensor. This is the key point of TDM in three dimensions since magnitude of this norm depends on the coordinate system. In two dimensions, the axis of a vortex is always perpendicular to the flow plane and thus the values of residual tensor and residual vorticity can be easily obtained analytically. Since the effective shear in 3D is given by the maximum value of the respective term (see below) over all possible rotations of the frame, the appropriate reference frame in which the maximum is obtained, called Basic Reference Frame (BRF), must be identified. Introducing an orthogonal transformation matrix Q, the value of ||∇u|| remains the same under all rotations Q(∇u)QT and using standard strain-rate and vorticity tensors S and , the expression residual 2 2 tensor + 4(|S12 Ω12 | + |S23 Ω23 | + |S31 Ω31 |) = ||∇u|| ,

(3)

holds in each reference frame in three dimensions, see [5]. Clearly, minimum of the residual tensor norm corresponds to the maximum of the term |S12 Ω12 | + |S23 Ω23 | + |S31 Ω31 | ,

(4)

over all possible frames, which offers a computational tool for extraction of the effective shear. The residual vorticity tensor is simply given by the skew-symmetric part of the residual tensor. Since BRF is in general different and independent for each point of the flow field in 3D, whole range of angles must be taken into account at every position through the transformation matrix Q, see [5]. Naturally, for practical calculations these ranges must be discretized uniformly with reasonable step size representing rotation with respect to a given axis. Obviously, finer step size allows better localization of the BRF while it results in an increase of required objective function evaluations since there is O(1/h3 ) dependence on the step size h.

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3 Numerical Tests To prove feasibility and efficiency of TDM, numerical tests have been carried out both in two and three dimensions.

3.1 Two-Dimensional Problems In two dimensions, residual vorticity can be shown to correspond directly to rigidbody rotation. Calculations are simplified by the fact that the BRF and relevant quantities can be easily obtained analytically. Thus, application of TDM is straightforward and the results readily explainable in physical terms. The method has been applied to vortex identification in a flow of incompressible fluid described by Navier-Stokes equations past NACA 0012 airfoil. Flow data have been obtained by FEM solver using Taylor-Hood elements with quadratic approximation of velocity and linear pressure approximation, satisfying Babuška-Brezzi condition. The solver is fully implicit with backward Euler difference for time derivative. The resulting system of non-linear equations is solved by Newton method and frontal method modified by Hood and Taylor for nonsymmetric matrices is applied for solution of the linearized system. For details concerning the solver and issues of extending its stability for high Reynolds numbers, see [2, 3, 7, 8]. The results presented below have been calculated at an angle of attack α = 34 deg and Reynolds number Re=100 so that stationary solution exists. Vortex as identified by TDM has been compared to vorticity evaluation obtained by double decomposition and represented by the two-dimensional vorticity tensor component ω (ω = (vx − u y )/2). The comparison is given in Figs. 1 and 2 and in detail in Figs. 3 and 4. It can be seen immediately that the vorticity as identified by ω is dominated by the shear component which obscures completely the vortex structures and no reasonable information can be extracted. Formation of these structures is

OMEGA 0.12 0.1 0.08 0.06 0.04 0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12

Fig. 1 Two-dimensional vorticity ω in a flow past NACA 0012 airfoil at α = 34◦ and Re = 100

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Fig. 2 Vorticity as identified by the residual vorticity (TDM) in a flow past NACA 0012 airfoil at α = 34◦ and Re = 100

Fig. 3 Detailed view of two-dimensional vorticity ω in a flow past NACA 0012 airfoil at α = 34◦ and Re = 100

clearly shown on the other hand by the residual vorticity as implied by TDM. In two dimensions, the TDM seems to offer a marked advantage.

3.2 Three-Dimensional Problems While in two dimensions TDM is very competitive, in three dimensions it poses a practical issue of computational costs due to the necessity of finding the BRF. Our experiments have shown that in general cases the choice of step of axis rotation might have a large impact on the quality of the identification and thus the step size of 1–5◦ should be recommended. Since BRF is specific for each point and therefore must be found for all nodes of the three-dimensional mesh, computational times


229

Fig. 4 Detailed view of vorticity as identified by the residual vorticity (TDM) in a flow past NACA 0012 airfoil at α = 34◦ and Re = 100

are very large for practical problems (2,981,251 calculations for one point with 1 deg step of rotation). To achieve reasonable times comparable to the most popular identification method, the λ2 -method, see [4], the procedure has been parallelized. The parallelization capitalizes on the fact that BRF for each point, and therefore also its calculation, is totally independent of other points. This has allowed the specific properties of graphical card calculations to be exploited using the CUDA programming environment. After parallelization, the computational times for TDM become comparable to those of λ2 -method. The DNS data for three-dimensional tests have been provided by the team of Prof. Rist, IAG Stuttgart, and represent a transient boundary layer, see [1], in which the typical -shaped vortex is formed. A vortex as identified by the residual vorticity is depicted in Fig. 5 and the results of λ2 -method are given in Fig. 6 for comparison.

Fig. 5 -shaped vortex as identified by the residual vorticity (TDM) in a transient boundary layer

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Fig. 6 Ω-shaped vortex as identified by λ2 -method in a transient boundary layer

The 3D results shown below hold for a threshold less than 10% of maximum value of corresponding quantities.

4 Conclusions The TDM employed in the paper appears to be competitive to well-established vortex-identification methods, namely the widely used λ2 -method, in terms of computational costs and quality of vortex identification. Being based on a natural decomposition of fluid motion into three parts, TDM offers a good intuitive insight into the process of vortex identification. The method is subject to ongoing research and some related schemes are being developed. Acknowledgements The authors are very grateful to Prof. Ulrich Rist and Dr. Kudret Baysal, IAG Stuttgart, for providing the three-dimensional DNS data used in the present chapter. This research has been supported by Grant Agency of Czech Academy of Sciences under grant IAA200600801 and by Czech Academy of Sciences through AV0Z20600510 and AV0Z10190503.

References 1. Bake, S., Meyer, D.G.W., Rist, U.: Turbulence mechanism in Klebanoff transition: a quantitative comparison of experiment and direct numerical simulation. J. Fluid Mech. 459, 217–243 (2002) 2. Burda, P., Novotný, J., Sousedík, B.: A posteriori error estimates applied to flow in a channel with corners. Math. Comp. Simulation 61, 375–383 (2003) 3. Burda, P., Novotný, J., Šístek, J.: On a modification of GLS stabilized FEM for solving incompressible viscous flows. Int. J. Numer. Meth. Fluids. 51, 1001–1016 (2006) 4. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995) 5. Koláˇr, V.: Vortex identification: new requirements and limitations. Int. J. Heat Fluid Flow. 28(4), 638–652 (2007)


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6. Koláˇr, V., Savory, E.: Dominant flow features of twin jets and plumes in crossflow. J. Wind Eng. Ind. Aerodyn. 95, 1199–1215 (2007) 7. Moses, P., Burda, P., Novotný, J.: Numerical solution of fluid - structure interaction by FEM. In: Neittaanmaki, P., Kˇrížek, M. (eds.) Finite Element Methods, Three-Dimensional Problems, GAKUTO Internat. Series, Mathematical Sciences and Applications 15, pp. 183–195. Gakkotosho, Tokyo (2001) 8. Moses, P., Novotný, J., Burda, P.: Finite element method in domain-decomposition for fluidstructure interaction problems. In: Groth, C., Zingg, D.W. (eds.) Computational Fluid Dynamics 2004, pp. 851–852. Springer, Berlin (2006)

Part IX

Design Optimization

Multi-point Optimization of Wind Turbine Blades Using Helicoidal Vortex Model Marcel Wijnen and Jean-Jacques Chattot

Abstract The availability of thorough wind assessment data has raised the question whether it would be possible to improve wind turbine power outputs with respect to existing data. At the University of California Davis wind turbine analysis codes have been developed as well as a wind turbine optimization code. This optimization is performed with respect to a specified wind speed. This chapter describes an attempt to expand the single-point optimization, that is performed with respect to one wind speed, to a multi-point optimization in which the probability distribution of wind speeds is taken into account. This multi-point optimization is expected to show the ability to trade-off between performances at different wind speeds. The trade-off is based on power output and the chance of occurrence. An objective function that describes the power production with respect to the discretized distribution function is defined. The optimization of the objective function is performed with respect to a limited drag force on the supporting tower.

1 Introduction In modern wind power engineering more sophisticated methods are used for wind assessments and modeling wind resources in complex areas. This has led to better strategies for planning wind turbine locations. For investors wind assessments have played a critical role, since those assessments best indicate expected revenues. The availability of wind assessment data has led to the question whether it would be possible to improve the design of a turbine blade by using this information. Wind turbines are designed to operate at a certain wind speed, for off-design conditions the turbine operates less efficiently. This is where potentially more energy can be harvested. Several methods are being used for turbine analysis and optimizations. Multiple research groups are working on methods to solve the full Navier-Stokes (NS) M. Wijnen (B) Faculty of Applied Physics, Delft University of Technology, 2628 CJ Delft, The Netherlands e-mail: [email protected]


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equations. The work by Sorensen [8] provided the best results in a blind comparison with respect to the experiments performed by National Renewable Energy Laboratory (NREL)[5]. These experiments provide a well known benchmark for this type of calculations. Computationally more affordable is the hybrid method by Schmitz and Chattot [7]. This method combines a NS solver in the near field with a helicoidal vortex model in the far field. This prevents dissipation of the wake structure of which most NS solvers are know to suffer. The NS solver in the near field is able to recover detailed flow characteristics at the blade and the smaller NS region results in a tremendous speedup of the solution process. Still for the purpose of a multi-point optimization a hybrid method is computationally too demanding. Starting point for this investigation is the existing analysis and optimization codes that are developed by Chattot [1–3]. This work is based on the helicoidal vortex model (HVM) as proposed by Goldstein [6]. In the HVM the flow is modeled by a discretized vortex sheet leaving from the trailing edge of the turbine blade into the wake towards infinity. The vortex strength Γ is related to the lift forces at the blade. From the vortex sheet the flow field can be calculated using the Biot-Savart law. The flow field determines the aerodynamic forces at the blade. This interdependency can be solved by iteration after which the flow field and the forces on the blade are fully described by Γ . A detailed description of the model can be found in [1–3]. The objective is to design a turbine blade that provides maximum power output with respect to the wind speed distribution function. An algorithm is developed that allows for the trade-off between turbine performances at different wind speeds. The optimization described in [3] is performed with respect to a single wind speed. This single-point optimization does not have the ability to trade-off between performances in off-design conditions. Changing the blade will have its effect on turbine performance at low, medium and high speed. The trade-off is made by accounting for the amount of energy that is produced at a certain wind speed and the probability with which that wind speed occurs. The objective function to be optimized is defined as the sum over the power outputs at the discretized wind speeds multiplied by their change of occurrence. For each wind speed the same operating strategy or constraint is applied. The maximum allowed drag force, that is used as a constraint, results from a structural analysis that is performed by the producer of the wind turbine.

2 Optimization The ultimate goal is to design the most efficient blade taking into account multiple wind speeds i and the probability with which they occur pi . The greatest step in doing that is adding a second operating wind speed to the single-point optimization. The variables of interest are the power coefficient C Pi (Γ i ) and the drag coefficient C Di (Γ i ). The variable that has to be optimized is the total power output CP =

2 i=1

pi C Pi .

(1)

237

z−direction

z−direction

Multi-point Optimization of Wind Turbine Blades Using Helicoidal Vortex Model

y−direction

y−direction

x−direction

x−direction

Fig. 1 Operation at lower and higher wind speeds. Operation at both wind speeds is constrained by equal drag force

The optimization is performed with the method of Lagrange multipliers. In the twopoint case i = 1 is used for the lower wind speed and i = 2 for the higher wind speed. For both wind speeds the following constraint is applied. C Di ≤ C Dtarget,i

(2)

In Fig. 1, operation of the same turbine at two different wind speeds is depicted. Multiple wind speeds will result in multiple vortex strength distributions Γ i . Therefore, it is more convenient to choose the chord c and twist t distributions as optimization variables. This provides Γ i (c, t, βi ), where βi refers to the pitch angle which will be discussed later on. The value of C P depends on the blade geometry and the pitch angles only. C P (Γ 1 , Γ 2 ) = C P (c, t, β1 , β2 )

(3)

This change of variables is required, since it is impossible during a solution process to correctly adjust the set of {Γ 1 , Γ 2 } which is implicitly related through the HVM. Adjusting the set {c, t, β1 , β2 } does not provide any problems. Performing the optimization with respect to the blade geometry is natural. The Lagrange multiplier optimization for this problem is given by 2

pi

i=1

j x−1 k=2

j x−1 2 ∂C P ∂Γk ∂C D ∂Γk + λi =0 ∂Γk ∂tl ∂Γk ∂tl

(4)

j x−1 2 ∂C P ∂Γk ∂C D ∂Γk + λi = 0, ∂Γk ∂cl ∂Γk ∂cl

(5)

i=1

k=2

and 2 i=1

pi

j x−1 k=2

i=1

k=2

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where j x is the number of blade sections. In order to simplify the notation the indices i are left out for C P , C D and Γk . Implementing variable pitch in the optimization guarantees that the constraints can be met for all operating wind speeds. Variable pitch is the possibility to rotate the blade around its long axis. This plays great part in the turbine operating strategy. Setting the pitch angle for i = 2 to zero leaves only β1 as variable pitch angle. To simplify notation the index of β1 will be left out. For the two-point case the pitch equations simplify to a single equation. p1

∂C P ∂C D + λ1 =0 ∂β ∂β

(6)

For this two-point optimization the rotational speed is taken to be constant for both wind speeds. It is straight forward to add it to the optimization in the future. Also, a more detailed operating strategy can be implemented by changing the constraints. At this point the chosen setup is ideal for the comparison between the single- and multi-point optimizations. To simplify the discussion on multiple Lagrange multipliers, the function f which represents the function that has to be optimized is introduced. For the representation of the constraints at both wind speeds g1 and g2 are introduced. The notation of the optimization formulation therefore changes into ∇ f + λ1 ∇g1 + λ2 ∇g2 = 0.

(7)

In a d-dimensional space the two gradient vectors ∇g1 and ∇g2 span a twodimensional plane. The plane is described by the following equation. x = λ1 ∇g1 + λ2 ∇g2

(8)

A solution to the optimization equation is found when vector ∇ f lies in x. At that point the Lagrange multipliers are such that the sum of the constraint gradient vectors match ∇ f . The solution is found as follows. The first objective is to keep the solution on the constraint surface and the second is to minimize the error in the optimization equations. Starting with an arbitrary solution, c, t and β will be adjusted to fit the constraints. The optimization variables collectively will be referred to as y. Once the constraints are met the initial solution lies on the constraint surface. In order to keep the solution on that surface, movements perpendicular to the surface are prohibited. This is stated as follows. Δy · ∇gi = 0

,

i = 1, 2

(9)

Desired movement is in the direction of ∇ f , but constrained in the directions as stated above. The Gram-Schmidt procedure can be used to subtract the directions in ∇ f perpendicular to the constraint surfaces. The Gram-Schmidt procedure is

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designed to create a set of orthogonal vectors starting with a set of arbitrary linear independent vectors. Applying this method allows to calculate the component of ∇ f that is perpendicular to ∇g1 and ∇g2 . The vectors ∇g1 and ∇g2 are strongly aligned. The constraint gradient vectors span a plane in which any movement is prohibited. There is an advantage in identifying this plane, since it allows a change in basis vectors that span the plane. Any set of two vectors that span the same plane can be used to constrain the movement towards the next solution. Exploiting this possibility to define two orthonormal vectors greatly simplifies the calculation of the Lagrange multipliers. Orthogonality results in a simple way to calculate the Lagrange multipliers by taking the inner product and normality avoids great differences in size of the vectors. From this point ∇g1 and ∇g2 refer to the new set of orthonormal vectors that span the plane of prohibited movements. The Lagrange formulation is stated to be Δy = η(∇ f + λ1 ∇g1 + λ2 ∇g2 ) λ1 = −∇g1 · ∇ f λ2 = −∇g2 · ∇ f.

(10) (11) (12)

The main advantage of this formulation is the fact that it is suited for generalization to more than two dimensions which allows it to be used for the multi-point optimization.

3 Results and Discussion In order to compare the single- and two-point optimization their overall performance is compared. The maximum overall performance is calculated from the optimum designed blades at both wind speeds. The results of this comparison are depicted in Fig. 2. Indeed, the two-point optimization finds a blade geometry that outperforms the best single-point designed blade. Unfortunately, the gain in power output is marginal and vanishes compared to the possible 1.5[%] gain as drawn in Fig. 2. Although the gain is small, it is important that the two-point optimization does find the highest power output. The geometry of the blade found with the two-point optimization is very distinct compared to its single-point counterpart. From this it can be concluded that a different blade is designed with slightly better performance. This suggests that the code might be useful for designing turbine blade geometries that are more cost effective to produce while remaining competitive on overall performance. This can be done by introducing a term related to the complexity of the geometry and implement that in the optimization functional. The main disadvantage of this method is the fact that, uniqueness of the solution is not guaranteed. The algorithm finds local optima only and therefore does not exclude the possibility that a blade geometry with more desirable performances exists.

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Loss [%]

4 3 2 1 0

10

12 14 16 18 Design wind speed SPO [m/s]

20

Fig. 2 Loss in [%] with respect to maximum overall power output as function of the single-point optimization (SPO) design speed. The dotted line renders the performance of the blade found with the two-point optimization. For this experiment p1 = 0.75 and p2 = 0.25

The main advantage of the method is its flexibility and the fact that the setup of the optimization allows for implementation of multiple wind speeds and different operating strategies.

References 1. Chattot, J.-J.: Design and analysis of wind turbines using helicoidal vortex model. Jpn. Soc. CFD, CFD J. 11(1), 50–54 (2000) 2. Chattot, J.-J.: Optimization of propellers using helicoidal vortex model. Jpn. Soc. CFD, CFD J. 10(4), 429–438 (2001) 3. Chattot, J.-J.: Optimization of Wind Turbines Using Helicoidal Vortex Model. The American institute for aeronautics and astronautics AIAA Paper No. 2003-0522, Reston, USA (2003) 4. Duque, E.P.N., Burklund, M.D., Johnson, W.: Navier-Stokes and comprehensive analysis performance predictions of the NREL phase VI experiment. J. Solar Energy Eng. 125(4), 457–467 (November 2003) 5. Fingerish, L., Simms, D., Hand, M., Jager, D., Cortrell, J., Robinson, M., Schreck, S., Larwood, S.: Wind Tunnel Testing of NREL’s Unsteady Aerodynamics Experiment. AIAA Paper, No. 2001-0035, Reston, USA (2001) 6. Goldstein, S.: On the Vortex Theory of Screw Propeller. The Royal Society, 123(792), 440–465 (April 6th 1929) 7. Schmitz,S., Chattot, J.-J.: A parallelized coupled navier-stokes/vortex-panel solver. J. Solar Energy Eng., Am. Soc. Mech. Eng. 127(4), 475-487 (November 2005) 8. Sorensen, N.N.: Navier-Stokes Predictions of the NREL Phase VI Rotor in the NASA Ames 80 ft × 120 ft Wind Tunnel. Wind Energy. Special Issue: Analysis and Modeling of the NREL Full-Scale Wind Tunnel Experiment, 5(2–3), 251–169 (2002)

Performance Evaluation of the Numerical Flux Jacobians in Flow Solution and Aerodynamic Design Optimization Alper Ezertas and Sinan Eyi

Abstract A direct sparse matrix solver is utilized for the flow solution and the analytical sensitivity analysis. The effects of the accuracy of the numerical Jacobians on the accuracy of sensitivity analysis and on the performance of the Newton’s method flow solver are analyzed in detail. The gradient based aerodynamic design optimization is employed to demonstrate those effects.

1 Introduction The Flux Jacobian matrices, the elements of which are the derivatives of the flux vectors with respect to the flow variables, are needed to be evaluated in implicit flow solutions and in analytical sensitivity evaluation methods. The main motivation behind this study is to explore the accuracy of the numerically evaluated flux Jacobian and the effects of the errors in those matrices on the convergence of the flow solver, on the accuracy of the sensitivities and on the performance of the design optimization cycle. To perform these objectives an exact Newton’s method flow solver is developed for the Euler flow equations. Flux Jacobian is evaluated both numerically and analytically for different upwind flux discretization schemes with second order MUSCL [6] face interpolation. The UMFPACK [1], unsymmetrical multifrontal sparse matrix solver is utilized to factorize the Jacobian.

2 Accuracy of Numerical Jacobians Numerical Jacobian matrices are derived by the first order finite differencing [2] R(w + en ε) − R(w) ∂R = ∂w ε

(1)

S. Eyi (B) Middle East Technical University, 06531 Ankara, Turkey e-mail: [email protected]


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where R(w) is the residual vector, w is the flow variables vector, en is the unit vector and ε is the finite difference perturbation magnitude. The norm of the differences between the elements of the numerical and the analytical Jacobian matrices is defined as the error. Numerical flux Jacobian matrices are derived with wide range of finite difference perturbation magnitudes and they are compared with analytically derived one. The optimum perturbation magnitude, which minimizes the error in the numerical evaluation, is searched. The found optimum value is compared with the value that is proposed by the well known simple formula which only requires the machine epsilon information [3] 6 εOPTIMUM = 2 !machine

(2)

Using the above formula the optimum perturbation magnitude can be estimated as 1.0 × 10−8 in double precision computations. The success of the estimation of the optimum is examined for different cases where the geometry, flow regime, discretization technique and the mesh resolution were varied. For all cases the simple formula approximated the optimum finite difference perturbation magnitude satisfactorily as it can be seen in Figs. 1 and 2. The effect of the limiters, which are used in the high order discretization, on the accuracy of the Jacobian matrix is also examined. Previously it was shown that utilization of limiters results in convergence problems for steady solutions due to the reaction of the limiters to even very small oscillations in smooth regions of the flow domain which introduce high non-linearity. Van Albada and Venkatakrishnan tuned the continuous limiter functions to prevent the activation of the limiter in smooth regions by introducing an additional parameter [5, 7]. The effect of the parameter on the accuracy of numerical Jacobians is given in Fig. 3. The usage of very small value

||Relative Error in Numerical Jacobian||

101

L1 , single L∞ , single LF , single L1 , double L∞ , double LF , double estimated, double estimated, single

100 10–1 10–2 10–3 10–4 10–5 10–6 10–7 10–8 –14

10

–12

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–10

10

–8

10

–6

ε

10

–4

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–2

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0

10

Fig. 1 The variation of the accuracy of the numerical Jacobian matrix with respect to the finite difference perturbation magnitude

||Relative Error in Numerical Jacobian||

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1st order Steger Warming 1st order Van Leer 1st order AU SM 2nd order Steger Warming 2nd order Van Leer 2nd order AUSM

10–1 10–2 10–3 10–4 10–5 10–6 10–7 10–8 –14

10

–12

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–10

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–8

–6

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ε

–4

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–2

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|| Relative Error in Numerical Jacobian ||F

Fig. 2 Effect of the discretization methods on the relative numerical Jacobian error 1st order ∈ = (cellarea)1/2 ∈ = 10 –3 ∈ = 10 –6 ∈ = 10 –9 ∈ = 10 –12

10–1 10–2 10–3 10–4 10–5 10–6 10–7 10–8 –14

10

–12

10

–10

10

–8

10

–6

10

–4

10

–2

10

ε

Fig. 3 The effect of the parameter ε on the relative numerical Jacobian error

significantly enlarged the magnitude of the second derivatives of the flux vector and as a result optimum perturbation magnitude is shifted and the total error is increased.

3 Convergence of the Newton Method Flow Solver A good initial guess is required to be able to obtain quadratic convergence rate from the Newton’s method. To strengthen the initial guess, time like terms are added to the diagonal of the Jacobian matrix in initial iterations. The initial and the withdrawal values of those added diagonal terms are found to be most critical factors on the convergence of the solution. Another factor that directly affects the convergence is the

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|| density residual || 2

10

ε = 10 –4 ε = 10 –6 ε = 10 –8 ε = 10 –10 ε = 10 –12

–6

10

analytic 2

–1

10

5

10 iteration #

15

20

Fig. 4 Residual history of the flow solution over NACA0012 airfoil at M=0.85, α= 1o

consistency between the schemes used in the Jacobian and the residual evaluations. It is found that, if the linearization of the Jacobian is not performed exactly, the iterations required for the converged solution will increase drastically. Compared to effects of initial conditioning and the exactness of the linearization, the effect of numerical Jacobian accuracy on the convergence is found to be negligible. The results of the study showed that the convergence of the Newton’s method is insensitive to the errors in the numerical Jacobian. The quadratic convergence rates are achieved independently from the Jacobian evaluation technique as long as the initial conditioning and the linearization of the matrix are applied properly. A convergence history example of the solver is presented in Fig. 4. The analytical and numerical evaluation methods are also compared in terms of the CPU time spent in a Newton iteration in Fig. 5. 1st order analyticall jacobian evaluation 1st order numerical jacobian evaluation solution with 1st order jacobiann evaluation 2nd order analytical jacobian evaluation 2nd order numerical jacobian evaluation solution with 2nd order jacobian

CPU time (seconds)

600

400

200

0 0

5000

10000

15000

node number

Fig. 5 Change of CPU time spent in each Newton iteration with grid size

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4 Sensitivity Analysis The sensitivity derivatives are evaluated by direct-differentiation method with discrete approach [4]. The reuse of the LU factors of the flux Jacobian that are evaluated in the flow solution enhanced the efficiency of the sensitivity analysis significantly. To analyze Jacobian error effect on the accuracy of sensitivity evaluation, both of numerical and analytical Jacobians are used in sensitivity analyses and the results are compared. 7 7

7 7 ∂w 7 7 ∂R ∂w 7 7 7 ∂R 7 − − 7 7 ∂β numeric 7 ∂β ∂w numeric ∂w analytic 7 ∂R 7

7 analytic ≤ cond 7 7 7 ∂w 7 7 ∂R 7 ∂w 7 7 7 ∂w analytic 7 7 ∂β analytic 7

(3)

The upper bound of the relative error is estimated by the condition number of the Jacobian matrix. The actual relative error was found to be significantly smaller than that approximated bound. As it can be seen in Fig. 6, the accuracy of the sensitivities varies by the finite differencing perturbation magnitude and becomes maximum at the same optimum value previously mentioned. However, the relative error was found to be very small even for large and very small perturbation magnitudes.

5 Design Optimization

||Relative Errorin Numerical Sensitivity Vector||1

The inverse design optimization of the RAE 2822 is performed using the pressure distribution of the NACA0012 airfoil as the initial guess. During the optimization the pressure discrepancy between the design and target geometries are minimized by least square minimization AUSM, 1st order VL, 1st order S-W, 1st order AUSM, 2nd order VL, 2nd order S-W, 2nd order

100 10–1 10–2 10–3 10–4 10–5 10–6 10–7 –14

10

–12

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–10

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–8

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ε

–6

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–4

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–2

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Fig. 6 The variation of the accuracy of the sensitivities calculated by numerically evaluated Jacobian

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–2

10 10

–3

Convergence Parameter

10

–1

ε = 10 –4 ε = 10 –6 ε = 10 –8 ε = 10 –10 ε = 10 –12 analytical

0

5

10

designcycle

Fig. 7 Design convergence parameter history of the inverse design problem performed with analytical and numerical Jacobians

The effects of the sensitivity accuracy on the efficiency of the inverse design optimization were analyzed. Figure 7 presents the design convergence histories. Due to very small relative sensitivity error, the usage of both Jacobian evaluation methods resulted in identical designs in the same number of design cycles.

6 Conclusion The numerical evaluation of the Jacobian matrix can be very advantageous since the occurring relative error is very small and does not introduce any practical penalty in flow and sensitivity analysis. Moreover using the numerical evaluation, very complex schemes can be linearized effortlessly compared to the analytical one. The only penalty of the numerical Jacobian usage is the increase in the CPU time spent.

References 1. Davis, T. A.: UMFPACK Version 4.1 User Manual. University of Florida, Florida (2003) 2. Dennis, J.E., Schnabel, R.B.: Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice Hall, Englewood Cliffs, New Jersey (1983) 3. Gill, P.E., Murray W., Wright M.H.: Practical Optimization. Academic Press, London (1992) 4. Korivi V.M., Taylor A.C., Newman P.A., Hou G.J.W., Jones H.E.: An Approximately Factored Incremental Strategy For Calculating Consistent Discrete Aerodynamic Sensitivity Derivatives. AIAA Paper 92-4746 (1992) 5. Van Albada, G.D., Van Leer, B., Roberts, W.W.: A comparative study of computational methods in cosmic gas dynamics. Astronomy Astrophys. 108, 76–84 (1982) 6. Van Leer, B.: Towards the Ultimate Conservative Difference Scheme, V. A Second Order Sequel to Godunov’s Method. J. Comput. Phys. 32, 101–136 (1979) 7. Venkatakrishnan, V.: Preconditioned conjugate gradient methods for the compressible NavierStokes equations. AIAA J. 29, 1092–1100 (1991)

Design Optimization in Non-equilibrium Reacting Flows Sinan Eyi, Alper Ezertas, and Mine Yumusak

Abstract The objective of this study is to develop a reliable and efficient design tool that can be used in chemically reacting flows. The flow analysis is based on axisymmetric Euler and the finite rate reaction equations. These coupled equations are solved by using Newton’s method. Both numerical and analytical methods are used to calculate Jacobian matrices. Sensitivities are evaluated by using adjoint method. The performance of the optimization method is demonstrated for a rocket motor nozzle design.

1 Introduction The reliability of design methods depends on the accuracy of flow models used. In order to capture the chemically reacting and rotational flow physics, the finite-rate reaction and axisymmetric Euler equations are solved simultaneously. In gradient base design optimization, the derivatives of objective function with respect to design variables are needed. In literature, these derivatives are called sensitivities. The accurate and efficient calculation of sensitivities is important for the performance of design method. In design optimization, using implicit methods for flow analysis is advantageous because the evaluation of sensitivities is very efficient. In order to improve the efficiency of design method and to reduce the numerical stiffness that occurs in the solution of reaction equation, Newton’s solution method is used. Newton’s method needs the Jacobian matrix which is the derivative of residual vector with respect flow variable vector. Analytical or numerical methods can be used in the calculation of Jacobian matrices. The analytical method is more accurate and faster compare to the numerical method [4]. However, the implementation of numerical method is much easier; the analytical method requires code development. Direct differentiation or adjoint methods can be used to evaluate the sensitivities. Adjoint method has more advantageous because the Jacobian matrix is

S. Eyi (B) Middle East Technical University, 06531 Ankara, Turkey e-mail: [email protected]


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solved only once to evaluate the sensitivities of all design variables. In order to avoid solving large Jacobian matrix for each design variable the adjoint method is used.

2 Reacting Flow Solver The formulation is based on the conservation equations of mass, momentum, energy and species concentrations for a chemically reacting system of Ns species R(Q) =

∂E 1 ∂(y n G) + −S=0, ∂x y ∂y

(1)

where n = 0 for two dimensional and n = 1 for axisymmetric coordinates, respectively. The flow variable vector, Q, and convective flux vectors E and G in the x and y directions. The fluxes are computed by Van-Leer upwinding schemes, and the second order discretizations are implemented. The source term vector S, contains contributions from chemical reactions. The chemical reaction includes 8 species. They are CO2 , CO, OH, H, O2 , H2 , H2 O, O. The reactions considered are given in Table 1. ⎞ ⎛ ⎞ ⎛ ⎞ ⎞ ⎛ ρi v ρi u ρi ω˙ i ⎜ ρuv ⎟ ⎜ ρu ⎟ ⎜0 ⎟ ⎜ P + ρu 2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ Q=⎜ ⎝ ρv ⎠ E = ⎝ ρuv ⎠ G = ⎝ P + ρv 2 ⎠ S = ⎝ 0 ⎠ i = 1, 8 (2) ρE 0 ρu H ρv H ⎛

In Newton’s method, the change in flow variable vector at the nth iteration can be calculated as:

∂R ∂Q

n

ΔQ n = −R Q n

(3)

Jacobian matrices are evaluated with both analytical and numerical methods [3]. The sparse Jacobian matrix is LU factorized and solution is executed by using UMFPACK sparse matrix solver [1]. The boundary conditions are implemented implicitly.

Table 1 Chemistry models for simulation Reactions OH + CO⇔H + CO2 CO + O2 ⇔CO2 + O O2 + H⇔O + OH

H2 + O⇔H + OH H2 O + O⇔2OH H2 + OH⇔2O + H

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3 Design Optimization In adjoint method, the Lagrangian function, L, is defined as: L = F (Q (Di ), X (Di ), Di ) + ΛT R (Q (Di ), X (Di ), Di ) 9: ; 8

(4)

=0

where F is the objective function, Di is the ith component of design variable vector, X is the coordinate vector of grid points, and Λ is the adjoint variable vector. Differentiating above equation with respect to design variable gives: dL ∂F ∂F dX + = + . d Di ∂ X d Di ∂ Di ⎫ ⎧ ⎪ ⎪ ⎪ ⎪ 0 is the mass density distribution and f denotes 2 2 the body forces. We have denoted by St = L/(Vc Tc ), Fr = V /(L f c ), the Strouhal 2 and Froude numbers and we introduce B = κc / ρc Vc , where ρc , Vc , L , κc , Tc are the characteristic density, velocity, length, stress and time respectively. Since we deal with an incompressible fluid, we get div u = 0

in D(t).

(2)

The conservation of mass becomes St

∂ρ + u · ∇ρ = 0 ∂t

in D(t).

(3)

If we denote by D(u) = (∇u + ∇ T u)/2 the rate of deformation tensor, the constitutive equation of the Bingham fluid can be written as follows: σ =

2η1 D(u) D(u) + η2 B if |D(u)| = 0, Re |D(u)| |σ | ≤ η2 B if |D(u)| = 0,

(4) (5)

where η1 ≥ η0 > 0 is the non dimensional viscosity distribution depending on ρ and η2 ≥ 0 is a non-negative continuous function which stands for the non dimensional yield limit distribution in D(t). Here Re = ρc Vc L/ηc is the Reynolds number and ηc is a characteristic viscosity. Note that if κc is the characteristic yield stress then B = Bi/Re, where Bi = κc L/(ηc Vc ) is the Bingham number.

Augmented Lagrangian/Well-Balanced Finite Volume Method

273

When considering a density-dependent model, the viscosity coefficient η1 and the yield limit η2 depend on the density ρ through two constitutive functions, i.e., η1 = η1 (ρ),

η2 = η2 (ρ).

(6)

In order to complete Eqs. (1, 2, 3, 4, 5, and 6) with the boundary conditions we assume that ∂D(t) is divided into two disjoint parts so that ∂D(t) = Γb (t) ∪ Γs (t). On the boundary Γb (t), which corresponds to the bottom part of the fluid, we consider a Navier condition with a friction coefficient α and a no-penetration condition : σt = −αu t , u · n = 0 on Γb . Here n stands for the outward unit normal on ∂D(t) and we have adopted the following notation for the tangential and normal decomposition of any velocity field u and any density of surface forces σ n : u = u n n + u t , with u n = u · n, σ n = σn n + σt with σn = σ n · n. The (unknown) boundary Γs (t) is a free surface, i.e. we assume a no-stress condition σ n = 0 on Γs (t), and the fact that the fluid region is advected by the flow, which can be expressed by St

∂1D (t) + u · ∇1D (t) = 0, ∂t

(7)

where 1D (t) is the characteristic function of the domain D(t). Finally the initial conditions are given by u|t=0 = u 0 , ρ|t=0 = ρ0 . The problem can be rewritten as a variational formulation (cf. Duvaut and Lions [5]) and we derive an asymptotic model in the shallow flow approximation. For sake of brevity, we have to refer to Bresch et al. [2] for the full 2D version. If we consider the 1D version of the obtained system, we have (where H is the height, V0 a height-averaged speed, is a test function and θ the slope angle): ∂(H V0 ) ∂H + = 0, ∂t ∂x ∂(ρ0 H ) ∂(H ρ0 V0 ) St = 0, + ∂t ∂x L 1 2

H ρ0 St∂t V0 ( − V0 ) + ∂x V0 ( − V0 ) d x 2 0 L L 4 βV0 ( − V0 )d x + H η1 (ρ0 )∂x (V0 ) + 0 0 Re L

√ BH η2 (ρ0 ) 2 |∂x ()| − |∂x (V0 )| d x ∂x ( − V0 )d x + St

(8) (9)

(10)

0

−1 L ≥ 2 H ρ0 sin θ ( − V0 ) Fr 0 L cos θ ε (H )2 ρ0 (∂x − ∂x V0 )d x. + 2 2 Fr 0 We further assume that ρ0 = cte, the model thus reduces to (8)–(10) but still contains the main difficulty.

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3 The Numerical Scheme We now describe a scheme which is able to compute stationary solutions. It mixes augmented Lagrangian and well-balanced (denoted WB, in the following) schemes philosophy. First, we make the semi-discretization in time:

St

L

∂(H n V0n ) H n+1 − H n + = 0, Δt ∂x

(11)

1

V0n+1 − V0n n+1 n+1 n 2 + ∂x V0 − V0 dx − V0 St Δt 2

H ρ0 n

0

L

+

0

βV0n+1 − V0n+1 d x

4 n H η1 (ρ0 )∂x V0n+1 ∂x − V0n+1 d x 0 Re L

√ BH n η2 (ρ0 ) 2 |∂x ()| − |∂x V0n+1 | d x + L

+

0

−1 ≥ 2 Fr +

ε Fr2

L

0 L

0

(12)

H n ρ0 sin θ − V0n+1

cos θ n 2 (H ) ρ0 ∂x − ∂x V0n+1 d x. 2

We now need to design a scheme for solving the variational inequality on one hand, and building the spatial discretization, on the other hand. For the inequality, we adopt an augmented Lagrangian method (see Dean et al. [4], Fortin and Glowinski [6]): (12) is equivalent to the minimization of a functional which is solved through the following procedure :

Step I.0 : We consider that we know V0n , H n , μn and q n . Then, we impose for k = 0, V k = V0n , μk = μn , q k = q n . Step I.1 : Compute q k+1 via : d k+1 = μk + r B(V k ) q

k+1

=

k+1

if d ≤ Bη2 d k+1 − Bη2 d k+1 if d k+1 > Bη2

0 1 r

Step I.2 : Compute V k+1 via :

d k+1

(13)


3 H

n

V k+1 − V0n ρ0 St + ∂x Δt

275

4 2 ρ0 V0n ε n = −βV k+1 + 2 H ρ0 cos θ 2 Fr

1 n H ρ0 sin θ + ∂x (H n (μk − rq k+1 )) Fr2 n 4η1 (ρ0 ) k+1 . + ∂x H + r ∂x V Re −

(14) Step I.3 : Update μk+1 via : μk+1 = μk + r (B(V k+1 ) − q k+1 ) Loop Steps I.1–I.3 : k → k + 1, until (e.g. with tol = 10−2 )

μk+1 −μk μk

≤ tol.

At convergence : We now have determined the value of V0 at time t n+1 , we just have to set : V0n+1 = V k+1 , and we also set μn+1 = μk , q n+1 = q k+1 . Note that if we look at the global problem, there is a coupling of the type : ⎧ ∂ H n V0n ⎪ H n+1 − H n ⎪ ⎪ + = 0, St ⎪ ⎪ Δt ∂x ⎪ ⎪ 3 2 4 ⎪ ⎪ ⎪ ρ0 V0n V k+1 − V0n ⎪ ε n n ⎪ ⎪ + ∂x + 2 H ρ0 cos θ = ⎨ H ρ0 St Δt 2 Fr 1 ⎪ ⎪ ⎪ −βV k+1 − 2 H n ρ0 sin θ + ∂x (H n (μk − rq k+1 )) ⎪ ⎪ ⎪ ⎪ Fr ⎪ ⎪ ⎪ ⎪ n 4η1 (ρ0 ) k+1 ⎪ + ∂x H + r ∂x V ⎪ ⎩ Re

(15)

which are shallow-water equations (SWE) in formulation(H, V ) with viscosity and source terms of various nature : natural ones coming from topography, and specific ones to present method, i.e. terms coming from the augmented Lagrangian. This is the reason that leads us to use well-balanced schemes (see [3]) to design the space discretization of present system. Again, for sake of brevity we have to refer to Bresch et al. [2] for the full details of the space discretization. The key point consists in treating Lagrangian variables μ and q in the same manner as in the well-balanced schemes for SWE, i.e. we propose a numerical flux which takes into account terms μ + r q as source terms, together with the topography terms. This leads to a coupling between height and velocity problems and to a scheme which is WB. This is illustrated in the following section (see Fig. 1), where a rectangular pulse is taken as initial condition. With the present scheme, the stationary solution is reached at t = 5 s. On the other hand, if we compare the evolution with two other schemes that do not make aforementioned coupling and do not take into account the Lagrangian terms, we see that the stationary state is not reached at t = 20 s.

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Fig. 1 Evolution of the material surface. Top: the solution with our WB scheme from t = 0 to t = 5 s where the stationary state is reached. Bottom: Comparison between the numerical result of the WB scheme (continuous thick line, actually the same simulation as on the Left) and two non WB schemes (dashed-dotted line and dashed line). These last two are still moving at t = 20 s whereas the WB one is sationary


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Acknowledgements D.B. thanks Institut de la Montagne (Université de Savoie) for financial support through the PPF: “Mathématiques et avalanches de neige, une rencontre possible?”. E.D.F.N. has been partially supported by the Spanish Government Research project MTM2006-01275 and the Région Rhône-Alpes (France). P.V. is partially supported by French ANR-08-JCJC-0104-01 research grant, as well as IMUS (Sevilla, Spain).

References 1. Balmforth, N.J., Craster, R.V., Sassi, R.: Shallow viscoplastic flow on an inclined plane. J. Fluid Mech. 470, 1–29 (2002) 2. Bresch, D., Fernández Nieto, E.D., Ionescu, I., Vigneaux, P.: Augmented Lagrangian Method and Compressible Visco-Plastic Flows : Applications to Shallow Dense Avalanches. Advances in Mathematical Fluid Mechanics, New Directions in Mathematical Fluid Mechanics, Birkhauser (Basel), pp. 57–89. doi: http://dx.doi.org/10.1007/978-3-0346-0152-8_4 (2010) 3. Castro Díaz, M.J., Chacón Rebollo, T., Fernández Nieto, E.D., Parés, C.: On well-balanced finite volume methods for nonconservative nonhomogeneous hyperbolic systems. SIAM J. Sci. Comp. 29, 1093–1126 (2007) 4. Dean, E., Glowinski, R., Guidoboni, G.: On the numerical simulation of Bingham visco-plastic flow: old and new results. J. Non Newton. Fluid Mech. 142, 36–62 (2007) 5. Duvaut, G., Lions, J.L.: Les inéquations en mécanique et en physique. Dunod, Paris (1972) 6. Fortin, M., Glowinski, R.: Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland Publ. Co., Amsterdam-New York (1983). 7. Gerbeau, J.-F., Perthame, B.: Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B. 1(1), 89–102 (2001)

Stable Simulation of Shallow-Water Waves by Block Advection Lai-Wai Tan and Vincent H. Chu

Abstract Waves in shallow water are computed using Lagrangian block hydrodynamics. The blocks transfer mass and momentum on a Lagrangian mesh. The computational stability is guaranteed by avoiding any usage of the Eulerian flux. The shock-capture and wave-front-tracking accuracies of the Lagrangian block hydrodynamics as determined from the convergence towards the exact solutions are approximately first order.

1 Introduction Computational stability is crucial to many water-engineering computation problems including the flooding over lands, the evolution of avalanches, and the wave run-up on beaches and overtopping of levees. The computation must capture the depth-andvelocity discontinuity across the shock waves and at the front as water advances and retreats over dry land. In the classical finite-volume methods, the simulations of these discontinuities often lead to unphysical numerical oscillations and subsequent breakdown of the computation. A variety of ad-hoc numerical methods have been introduced to manage the computational instability [6, 7, 9, 11, 20]. Shock capture schemes [5, 6] and flux limiters [11, 13] have been the methods to control numerical oscillations. The advance and the retreat of water on dry land have been attempted with some success using the wet-and-dry threshold [10, 12], the wet cell mapping [8], the artificial porosity [19] techniques, the volume-of-fluid method [7], the arbitrary Lagrangian – Eulerian algorithm [1], and the technique of the artificial viscosity [14, 20]. In the Lagrangian block hydrodynamics (LBH) method developed by Tan and Chu [16, 17], the mass and momentum are transferred by the Lagrangian advection of the blocks. This LBH method is absolutely stable. The generalization of the method to two dimensional problems are explained in this chapter by way of V.H. Chu (B) Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, QC, Canada e-mail: [email protected]


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a series of numerical simulations for convergence towards the exact solutions of Stoker [15] and Thacker [18]. The application of the method is demonstrated by the flood routing through meandering river.

2 Lagrangian Block Hydrodynamics A Lagrangian block is defined by its water depth h i,L j and the widths Δxi,L j = L L L L L xi+1, j − x i, j and Δyi, j = yi+1, j − yi, j of the blocks. Figure 1a shows the relative locations of the volume block to the momentum blocks on a staggered grid. At the beginning of the Lagrangian advection, at time t, the blocks fit the Eulerian mesh, that is xi,L j = xi, j and yi,L j = yi, j . At the end of the advection step, at time t + Δt, Δxi,L j Δyi,L j h i,L j = Δx i,L j Δyi,L j h i,L j for volume conservation. Figure 1b shows the advection of the volume block and Fig. 1c, d show the momentum blocks. The edge positions of the blocks xi,L j and yi,L j at time t + Δt are calculated by the momentum equations assuming hydrostatic pressure variation over the depth. To avoid the entanglement of the blocks, the blocks are reconstructed on the Lagrangian mesh to match the Eulerian mesh using the second-moment method [3, 4]. Stability of

Eulerian grid

Δx L

Eulerian

t+ Δ t

grid

Δy L

t u (i, j )

h (i, j )

Δy

Δy

u (i, j ) h (i, j ) v (i, j ) Δx

v (i, j ) Δx

(b) volume block

(a) staggered grid Δx L

Eulerian

t+ Δt

grid Δx L

Δy L

t u (i, j )

t+ Δ t Δy

v (i, j ) Δx

u (i, j ) t Δy

Eulerian grid

(c) x-momentum block

Δy L

v (i, j ) Δx

(d) y-momentum block

Fig. 1 (a) The staggered block system, (b) the volume block, (c, d) the x- and y-momentum blocks at the beginning and the end of a Lagrangian advection time step

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the LBH computation is guaranteed when the time step is selected to satisfy the Courant-Friedrichs-Lewy (CFL) condition.

3 Convergence to Exact Solutions The accuracy of the LBH method is assessed using the exact solutions of Stoker [15] and Thacker [18] as the benchmarks. Figure 2 shows the plan view of the simulated oblique shock waves in a 100 m × 100 m basin. The water depths are initially 10 and 1 m across a dam along the diagonal of the basin. Figure 3a, b show the cross sectional view of the depth and velocity profiles of the oblique shock wave at a time t = 2.5 s after the removal of the dam. Stoker [15] gave the height of the shock wave as h s / h o = 0.396. The convergence towards his exact solution as the block size is refined follows the first-order convergent law of Celik et al. [2] as shown in Fig. 3c, d. The computational stability of the LBH is absolute as the oblique shock waves are simulated without the interruption by any numerical instability. (d) t = 90 s

(c) t = 10 s

(b) t = 5 s

(a) t = 0.1 s ho = 10 m

hd = 1 m

Fig. 2 Plan view of the oblique shock wave depth at time t = 0.1, 5, 10 and 90 s

h ho

1

(hStoker − hs )

shock front

hStoker

0.5

0.01

0.001

hs ho = 0.396 0 −2

−1

p=1

0 1 x (t gho )

0.0001

2

0.00001 0.001 1

u gho 0.5

0 −2

(uStoker − us ) uStoker

−1

0

x (t gho )

1

2

0.01

0.1

0.01 0.001

0.0001 0.001

p=1 0.01 0.1 Δx (t gho )

Fig. 3 (a) Depth, and (b) velocity profiles across the shock wave at time t = 2.5 s. Circle symbol denotes the exact solution of Stoker [15]. Convergence of (c) depth, and (d) velocity as the block size is refined from Δx = 1–0.5, 0.2, 0.1, and 0.04 m

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(a)

Energy (MJoule/(kg/m3))

(b)

plan view

ΔE Kinetic Energy

5

0

A

A

Initial Energy Eo

10

5000

(c)

10000 15000 Time (s)

Potential Energy 20000

0. 01 T ΔE Eo Δt 0. 001 p=1

A - A section

h

zo

0.0001

1

10 Δx (ho)max

100

Fig. 4 (a) LBH simulation of the standing wave in the parabolic bowl using coarse block size, (b) kinetic plus potential energy of the water in the parabolic bowl with Δx = Δy = 5 m, (c) convergence toward the exact solution

The tracking of the wetting-and-drying interface is demonstrated by the LBH computation of the standing waves in a parabolic bowl as shown in Fig. 4a, b. Initiated by a mound of water, the water advances and retreats on the dry bed of the bowl as the water moves up and down under the influence of gravity. This is a difficult numerical problem. If the classical methods were used, the spurious numerical oscillations would produce negative water depth at the wave front leading to computation breakdown. However, the LBH computation is absolutely stable only on the CFL condition. Figure 4c shows the energy of the water obtained by the LBH computation over a period of 20 cycles of water sloshing in the parabolic bowl. The kinetic plus potential energy E is supposed to be constant according to the exact solution by Thacker [18]. Therefore, the energy loss T ΔE/Δt over one wave period is a measure of the computation error. The analysis of this error against the block size in Fig. 4d shows that the convergence of the LBH simulation towards the exact solution follows the first order ( p " 1.0) of Celik et al. [2].

4 Flood over Meandering River Numerous application examples have been conducted taking advantage of the computational stability of the LBH method [3, 4, 16, 17]. The meandering river shown in Fig. 5 is another application example. The river has a 10 m wide sinusoidal meander

Stable Simulation of Shallow-Water Waves by Block Advection 30 y (m)

ω (s−1) >2

(c) So = 0.01

(a) So = 0.0025

20 1

10 0

y (m)

283

0

(d) So = 0.02

(b) So = 0.005

20 −1 10 0

close to the leading-edge (left) and in the reattachment region (right) ()0.5

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Fig. √ 2 LES results (iso-lines) compared to Pprime lab. experiments (shaded isocontours): < v 2 > close to the leading-edge (left) and in the reattachment region (right)

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Fig. 3 LES results (iso-lines) compared to Pprime lab. experiments (shaded isocontours): < u v > close to the leading-edge (left) and in the reattachment region (right) 0.14 0 0.12

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Fig. 4 Streamwise wall pressure distributions (LES solid lines) compared to experiments from [4] (dots), on both the mean (left) and the fluctuating (right) pressure coefficients

in the separated flow. These points must be clarified in future works. While small discrepancies could be seen on the present statistical fields, we will see that main spectral flow arrangements can however be well described by the present LES. Spectral analysis: separated–reattached flows are characterized by two basic groups of frequency modes which are related to shedding and flapping phenomena. The vortex shedding resulting from the large scale motion of the mixing layer, is characterized by a frequency peak band around f L R /U∞ = 0.6−0.8 (called the shedding modes) [4, 10]. The flapping phenomenon is an overall dynamical mechanism links to successive enlargements and shrinkages of the separated zone. Its characteristic frequencies (namely the flapping modes) are much lower than the shedding modes, e.g. f L R /U∞ " 0.12 [4, 10]. These characteristic dimensionless frequencies are clearly visible on the energy spectra of the longitudinal velocity recorded by LES from probes located above the separated region (Fig. 5, left) where the first group corresponds to the flapping modes and the second group is related to the shedding modes. These main dimensionless frequencies are also recorded when the probe is localized within the mixing layer (x = 0.3, y = 0.158) with however

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lower energies (Fig. 5, right). Moreover, a peak is also clearly visible around f L R /U0 = 3.4 (Fig. 5, right). In fact, this frequency can be attributed to the Kelvin-Helmholtz mode of the mixing layer edging the separation since the Strouhal number (Stω = f δω /Uc = 0.33) based on the local vorticity thickness (δω = (< Uhigh > − < Ulow >)/ max y ∇ < U >) of the mixing layer and the local convection velocity (Uc = (< Uhigh > + < Ulow >)/2) is in very good agreement with the value generally admitted, recorded experimentally by numerous authors [1, 7]. Downstream this characteristic frequency, the decaying rate of the energy fits with the well known −5/3 slope over slightly less than one decade. The distribution of δω along the main flow direction agrees very well with the classical slope admitting an averaged value < dδω /d x >= 0.17, value recovered by Cherry et al. [4] through a collection of measurements resulting from the literature. Moreover, for single-stream mixing layer (i.e. with effectively zero velocity on one side) the value of the growth rate of δω generally admitted is in between 0.145 and 0.22 [2]. As pressure – velocity coupling is a key point in aeroacoustic prediction, 1 (x = 0.61; y = 0.61) (x = 1.52; y = 0.61)

0,25

slope :–5/3

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Eu

Eu

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(x = 1.52 ; y = 0.61) (x = 0.30; y = 0.158)

1e-05

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1

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f.Lr/U0

Fig. 5 Energy spectra of the fluctuating longitudinal velocity recorded by LES from probes located above the separated region and at the separated region border within the mixing layer. On the left: linear energy scale; on the right: logarithmic energy scale

slope ∼ 0.173

(x = 1.52 ; y = 0.61) (x = 0.30 ; y = 0.158)

0,25

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0,125 slope ∼ 0.228

0,1

Ep

max (U(y))/(du/dy)max

0,175

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0,025 0 0

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(x-x0)/Lr

0,8

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we also performed energy spectrum of the static pressure (Fig. 6, right). It comes out that the pressure seems to be strongly coupled to the velocity since flapping, shedding and Kelvin-Helmholtz modes are also displayed on pressure spectra (Fig. 6, right).

4 Conclusion LES results obtained with a high-order scheme coupled with a dynamical Smagorinsky subgrid model, favorably compare to experiments on the separated–reattached flows over a blunt flat plate. The main flow features are both qualitatively and quantitatively well predicted, namely the shedding, the flapping modes, the growth of the turbulent mixing layer edging the separation, and the statistical fields when coordinates are re-scaled by using the reattachment length (L R ). However, some small discrepancies are noticeable, mainly on the reattachment length, that has to be linked to the premature turbulence growth. This might be attributed to the influence of the grid spacing associated to the subgrid modeling. These relevant points need to be investigated. Moreover, we suspect that the low Reynolds number used in the present simulations could also affect the transition process and the turbulence development within the mixing layer which could also contribute to discrepancies encountered here. In the near future, sensitivity to the Reynolds number will be checked. Acknowledgements Authors would like to greatly acknowledge experimentalists from LEAPoitiers who recorded experimental data, for their valuable comments and discussions on data. This study has received financial support from the Agence National de la Recherche (ANR) through the DIB project. Authors also greatly acknowledge the support of the CNRS’s national supercomputing center (IDRIS) where part of computations have been carried out.

References 1. Bernal, L.P., Roshko, A.: Streamwise vortex structure in plane mixing layers. J. Fluid Mech. 170, 499–525 (1986) 2. Brown, G.L., Roshko, A.: On density effects and large structure in turbulent mixing layers. J. Fluid Mech. 64, 775–816. part 4. (1974) 3. Castro, I.P., Epik, E.: Boundary layer development after a separated region. J. Fluid Mech. 374, 91–116 (1998) 4. Cherry, N., Hillier, R., Latour, M.E.M.: Unsteady measurements in a separated and reattaching flow. J. Fluid Mech. 11, 13–46 (1984) 5. Daru, V., Tenaud, C.: High order one-step monotonicity preserving schemes for unsteady flow calculations. J. Comput. Phys. 193, 563–594 (2004) 6. Daru, V., Tenaud, C.: Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme. Comput. Fluids 38, 664–676 (2009) 7. Delville, J.: Characterization of the organization in shear layers via the proper orthogonal decomposition. Appl. Sci. Res. 53, 263–281 (1994)

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8. Djilali, N., Gartshore, I.S.: Turbulent flow around a bluff rectangular plate part i: experimental investigation. ASME J. Fluid Eng. 113, 51–59 (1991) 9. Hoarau, C., Bore, J., Laumonier, J., abd Gervais, Y.: Analysis of the wall pressure trace downstream of a separated region using extended proper orthogonal decomposition. Phys. Fluids 18, 055107 (2006) 10. Kiya, M., Sasaki, K.: Structure of large scale vortices and unsteady reverse flow in the reattaching zone of a turbulent separation bubble. J. Fluid Mech. 154, 463–491 (1985) 11. Yang, Z., Abdalla, I.E.: Effects of free-stream turbulence on a transitional separated-reattached flow over a flat plate with a sharp leading edge. Int. J. Heat Fluid Flow 30(5), 1026–1035 (2009)

Progress in DES for Wall-Modelled LES of Complex Internal Flows Charles Mockett and Frank Thiele

Abstract Very good agreement with benchmark data is achieved for IDDES of separating/reattaching flow over periodic hills. The effects of grid and time step coarsening are investigated, indicating a high degree of robustness of the method for this flow. A combination of IDDES with an all-y + RANS wall function is also examined and results suggest that this further enhances robustness.

1 Introduction Since its conception [9], detached-eddy simulation (DES) has become arguably the most widepsread technique combining Reynolds-averaged Navier–Stokes (RANS) modelling and large-eddy simulation (LES) for the simulation of turbulent flows. The most recent revision of the method, namely the “improved delayed DES” (IDDES) formulation [7], represents a significant extension: Whereas natural applications of DES foresee complete modelling of the boundary layer using RANS, IDDES incorporates modifications to improve results when applied as a wallmodelled LES (WMLES). WMLES targets resolution of the large-scale turbulent structures in the outer boundary layer using LES, with only the very near wall region modelled using RANS. This strategy is generally more appropriate for flows with thick boundary layers and weak separation (e.g. internal flows) and significantly reduces the computational cost scaling with Reynolds number (Re) relative to pure LES of wall-bounded flows. Although WMLES of channel flow using the original version of DES [4] was successful in some key respects, a shortcoming was identified in the region near the RANS-LES interface, where an upwards shift in the mean velocity profile was observed. This “logarithmic-layer mismatch” (LLM) gave rise to under-prediction of the skin friction coefficient of around 15%. The improvements achieved by the C. Mockett (B) Institute of Fluid Mechanics and Engineering Acoustics, Technische Universität Berlin, 10623 Berlin, Germany e-mail: [email protected]


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IDDES formulation were significant, with negligible LLM observed [7]. It was however acknowledged that further validation for more complex flows is necessary. This work pursues that objective by examining the performance of IDDES for flow involving separation and reattachment. In addition to an optimal setup, an assessment of the robustness of the approach to grid and time step coarsening is made. Finally, a combination of IDDES with an all-y + RANS wall function [6] is studied, with the goals of increasing robustness and assessing the importance of near-wall resolution.

2 Methodology All presented results were generated using the IDDES method [7] in combination with the Spalart – Allmaras RANS model [8]. The reduction of LLM for IDDES, described in Sect. 1, is achieved via a near-wall reduction of the LES filter width and a system of blending functions that significantly sharpen the RANS-LES interface and move it closer to the wall. The in-house CFD solver ELAN [11] is used, which is an implicit, pressurebased, cell-centred, multi-block structured, finite volume code of second order accuracy in space and time. Second order central differences are used for the convective fluxes and the solver is run in incompressible mode. An assessment of the numerics and code-specific calibration of the DES model constant has been conducted [3]. A hybrid wall boundary condition [6] is applied, which seamlessly blends a highReynolds log-law wall function with a low-Reynolds boundary condition such that grids with any wall cell spacing, y + , can be used.

3 Summary of Results for Channel Flow Fully-developed, turbulent channel flow has been computed on wall-refined grids (y + ≈ 1) for a wide range of Reynolds numbers. Due to space restrictions, these can only be summarised here, with reference to [3] for more detail. Good agreement with benchmark DNS and empirical correlations was seen. A slightly higher degree of LLM was seen in comparison with the results of the method authors [7], however the difference in C f was small to negligible.1 To provide an extreme test of the all-y + wall function and of the IDDES method,2 simulations using an equidistant Cartesian grid were carried out. For all Reynolds numbers over an extremely wide range, 590 ≤ Reτ ≤ 105 , good agreement of mean velocity and C f with benchmark data was achieved using the same grid.

1 The reasons for this discrepancy are not entirely clear, however we conjecture that a small degree of code-specific adjustment of the IDDES blending functions, which were empirically tuned for a fourth-order code, may alleviate this. 2 Many of the blending functions are designed for use with a wall-refined grid.

Progress in DES for Wall-Modelled LES of Complex Internal Flows

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4 Periodic, 2D Hills To provide increased complexity whilst maintaining reliable and directlycomparable benchmark data, the flow over periodic, 2D hills was chosen. A recent, highly-resolved LES [1] and experimental measurements [1, 5] are used for comparison. All quantities, including the Reynolds number, Reb , are normalised with respect to the hill height, h, and the bulk velocity at the hillcrest, Ub . Both data sources are available at Reb ≈ 10600, whereas only measurements are available at the higher Reb ≈ 37000. A wide range of IDDES have been performed, with variations of the grid, time step and Reb . These are summarised together with the chosen benchmark data in Table 1, which also lists global features of the flow.3 The computational domain was identical to the benchmark LES, namely L x = 9.0h, L y = 3.036h and L z = 4.5h.

4.1 Datum simulations The grid employed for the datum IDDES is shown in Fig. 1a. The high spanwise resolution ensures roughly cubic cells just downstream of the hill. The agreement with the benchmark LES at Reb = 10600 is excellent on all fronts, with the separation position predicted accurately and the reattachment slightly under-predicted (by 3%, Table 1). The C f predicted on the lower wall is in near perfect agreement (Fig. 2). Profiles of mean streamwise velocity, U , and Table 1 Summary of benchmark data and IDDES simulations of periodic, 2D hill flow Case name Reb (nom.) Ub (act.) dp/d x Nx Ny Nz Δt Tavg x s / h xr / h 10600-LES 10600-Exp 37000-Exp

10595 10600 37000

– – –

– – –

281 – –

222 – –

200 – –

0.0018 1277 – − – −

0.19 – –

4.69 4.21 3.76

10600-Datum 37000-Datum 10600-L2xyz 10600-L2z 10600-yEq 37000-yEq 10600-dt0.03 10600-dt0.06 10600-dt0.09 10600-dt0.18 10600-dt0.36

10595 37000 10595 10595 10595 37000 10595 10595 10595 10595 10595

0.991 0.992 0.977 0.972 1.024 0.993 0.970 0.967 0.966 0.901 0.871

−0.0106 −0.0091 −0.0106 −0.0106 −0.0106 −0.0091 −0.0106 −0.0106 −0.0106 −0.0106 −0.0106

160 160 80 160 160 160 160 160 160 160 160

160 160 80 160 78 78 160 160 160 160 160

120 120 60 60 120 120 120 120 120 120 120

0.015 0.015 0.030 0.015 0.015 0.015 0.030 0.060 0.090 0.018 0.036

0.19 0.26 0.23 0.17 0.45 0.53 0.19 0.19 0.20 0.21 0.25

4.55 4.21 4.70 4.60 3.82 2.91 4.27 4.33 4.30 4.26 4.26

3

1614 1053 1218 995 321 17 1233 1866 1044 2340 3654

The nominal Reb , the actual Ub , the imposed pressure gradient, dp/d x, the grid cell number in the streamwise, wall-normal and spanwise directions (N x , N y , N z respectively), the time step, Δt, the statistical time sample following elimination of initial transient, Tavg and the streamwise location of flow separation and reattachment, x s / h and xr / h are listed.

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Y

Z

Y X

Z

(a)

X

(b)

Fig. 1 x/y plane of computational grids (only every 2nd point shown). (a) Wall-refined datum grid, (b) equidistant wall-normal grid (“yEq”)

Reynolds shear stress, u v , are compared in Fig. 3 at x/ h = 0.5, an important location in the very early shear layer. U is predicted very well whereas the magnitude of u v is slightly over-predicted in the shear layer. This correlates with the slight under-prediction of xr . At Reb = 37000, the changes relative to the lower Re are qualitatively reproduced, although the deviation relative to experiments is slightly higher for xr (Table 1) and the mean velocity, and much higher for the Reynolds stresses.

4.2 Effect of grid coarsening Three different grid coarsening strategies have been investigated: The elimination of every second cell in the z direction (“L2z”), in all directions (with a corresponding doubling of Δt, “L2xyz”) and an equidistant grid in the y direction maintaining all other properties (“yEq”, Fig. 1b). The L2z results are closer to the datum than the L2xyz case in terms of C f (Fig. 2) and the L2xyz case also shows greater deviation from the U profile and the 0,030 10600-LES 10600-Datum 10600-L2z 10600-L2xyz 10600-yEq 10600-dt0.36

0,025 0,020

Cf

0,015 0,010 0,005 0,000 –0,005

0

1

2

3

4

5

6

7

8

9

x/h Fig. 2 Effect of alternative grid coarsening strategies and coarse time step on IDDES skin friction coefficient at lower wall in comparison with benchmark LES [1], Reb = 10600


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3 10600-LES 10600-Exp 10600-Datum 10600-L2z 10600-L2xyz 10600-yEq 10600-dt0.36

y/h

2,5

2

1,5

1 –0,2

0

0,2

0,4

0,6

0,8

1

U / Ub

1,2

–0,04

–0,03

–0,02

–0,01

0

2

u’v’ / Ub

Fig. 3 Effect of alternative grid coarsening strategies and coarse time step on IDDES mean velocity and resolved Reynolds shear stress profiles in comparison with benchmark LES [1] and experiment [1, 5], x/ h = 0.5, Reb = 10600

strongest under-prediction of shear layer turbulence (Fig. 3). The strongest deviation is given by the yEq grid despite good results using this strategy for channel flow (Sect. 3). Two principal explanations are offered for this: Firstly, the shallow early separation clearly cannot be resolved on this grid (see Fig. 3). Secondly, the wall function formulation assumes an equilibrium boundary layer, which is on average valid for the channel but far from valid for the periodic hills. The effect on xs , xr and the profiles is qualitatively very similar to that observed by Temmerman [10] in LES simulations with various wall functions and coarse wall-normal resolution. A very strong further degradation of the results was seen in that work for equivalent simulations omitting the wall functions. At Reb = 37000 the effect of the yEq grid relative to the datum grid appears to be similar to Reb = 10600, although the 37000-yEq results can only be considered preliminary due to a short statistical sample computed so far.

4.3 Effect of coarse time step Five additional simulations were conducted with various Δt, all coarser than the datum simulation, reaching up to a factor 24.4 As seen in Table 1, Ub (and hence the actual Reb ) decreases significantly as Δt increases, indicating a strong effect on the global flow prediction. Despite this, the effect on C f and the profiles (Figs. 2 and 3) is relatively mild.5 The effect of a coarse Δt on the mean eddy viscosity ratio, ν t /ν,

4 The computational cost is not however reduced proportionally, since the number of iterations required for convergence on each time step increases with Δt. 5 This led others to conclude that “the time step size has no influence on the averaged results” [2], however the practically-relevant relationship between dp/d x and Ub was not considered.

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is the opposite to that seen for grid coarsening: ν t /ν is reduced as Δt increases. The magnitude of the resolved Reynolds stresses increases with Δt.

5 Conclusions Very good agreement with benchmark LES is achieved by IDDES with significantly reduced computational expense (with an estimated factor 34 for the datum and 540 for the isotropically-coarsened cases) and clear improvement in solution quality with grid refinement is seen. Of the coarse grids, that with an equidistant wallnormal point distribution for the RANS wall function produced the lowest quality results, which is believed in part to reflect the importance of near-wall resolution for this case. It is thought that the all-y + BC nonetheless represents a robustness improvement,6 which is of significant practical importance. The effect of extreme time step coarsening on mean velocity, Reynolds stresses and skin friction was relatively benign, however a strong distortion of the mass flow for a specified pressure drop was seen. The effect of increased Reynolds number was quantitatively not well reproduced, which warrants closer attention. Although more complex than the channel, the periodic hills don’t necessarily represent a greater challenge for the method: Reasonable results are achieved despite severe violation of best practice and evidence is accumulating that the resolution of turbulent scales is in itself more important than modelling details. Nonetheless, the excellent results produced by IDDES reflect positively on the method. Further investigations of interest include the deactivation of the all-y + wall function and comparative calculations with earlier DES variants and alternative methods. Acknowledgements This work was funded by the EU DESider (AST3-CT-2003-502842) and ATAAC (ACP8-GA-2009-233710) projects with computing resources provided by the North German Supercomputing Alliance (HLRN, www.hlrn.de). The assistance of M. Fuchs is gratefully acknowledged. The chapter has benefitted from discussions with Dr. P. Spalart and Prof. M. Strelets.

References 1. Breuer, M., Peller, N., Rapp, C., Manhardt, M.: Flow over periodic hills – numerical and experimental study in a wide range of Reynolds numbers. Comput. Fluids 38, 433–457 (2009) 2. Kornhaas, M., Sternel, D., Schäfer, M.: Influence of time step size and convergence criteria on large eddy simulations with implicit time discretization. In: Meyers, J., et al. (eds.) Quality and Reability LES, pp. 119–130. Springer, The Netherlands (2008) 3. Mockett, C.: A Comprehensive Study of Detached-Eddy Simulation. PhD thesis, Technische Universität Berlin, Berlin. http://www.ub.tu-berlin.de/index.php?id$=$2375#c10583 (2009)

6

Similar investigations in the Literature [10] suggest that a far greater degradation of solution quality would arise without the wall function.


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4. Nikitin, N., Nicoud, F., Wasistho, B., Squires, K., Spalart, P.: An approach to wall modeling in large-eddy simulations. Phys. Fluids 12(7), 1629–1632 (2000) 5. Rapp, C.: Experimentelle Studie der turbulenten Strömung über periodische Hügel. PhD thesis, Technische Universität München, München (2009) 6. Rung, T., Lübcke, H., Thiele, F.: Universal wall-boundary conditions for turbulencetransport models. Z. angew. Math. Mech. 81(1), 1756–1758 (2000) 7. Shur, M., Spalart, P., Strelets, M., Travin, A.: A hybrid RANS-LES approach with delayed DES and wall-modeled LES capabilities. Int. J. Heat Fluid Flow 29(6), 1638–1649 (2008) 8. Spalart, P., Allmaras, S.: A one-equation turbulence model for aerodynamic flows. In: Proceedings of the 30th AIAA Aerospace Sciences Meeting and Exhibit, Reno (1992) 9. Spalart, P., Jou, W., Strelets, M., Allmaras, S.: Comments on the feasibility of LES for wings, and on a hybrid RANS/LES approach. Adv. DNS/LES, p. 1. Greyden Press, New York, NY (1997) 10. Temmerman, L.: Large Eddy Simulation of Separating Flows from Curved Surfaces. PhD thesis, Queen Mary University of London, London (2004) 11. Xue, L.: Entwicklung eines effizienten parallelen Lösungsalgorithmus zur dreidimensionalen Simulation komplexer turbulenter Strömungen. PhD thesis, Technische Universität Berlin, Berlin (1998)

Enhancement of the Performance of the Partial-Averaged Navier – Stokes Method by Using Scale-Adaptive Mesh Generation B. Basara and Z. Pavlovic

Abstract The Partially-Averaged Navier-Stokes (PANS) approach is a recently proposed method which changes seamlessly from Reynolds-Averaged NavierStokes (RANS) to the direct numerical solution of the Navier-Stokes equations (DNS) as the unresolved-to-total ratios of kinetic energy and dissipation are varied. The parameter which determines the unresolved-to-total kinetic energy ratio f k is defined based on the grid spacing. The PANS asymptotic behavior goes smoothly from RANS to DNS with decreasing f k . In the work of Basara, Krajnovic and Girimaji (Proceedings of ERCOFTAC 7th International Symposium on Engineering Turbulence Modelling and Measurements ETMM7, 2/3, pp. 548–554, Lymassol, Cyprus, 2008), it was shown that a dynamic update of the PANS key parameter f k by changing at each point and at the end of every time step, is the promising approach to provide the optimum modeling on employed computational meshes. This work is extended here by introducing the adaptive local grid refinement which keeps in advance prescribed value of the parameter f k . The results show benefits of using such advanced numerical technique in conjunction with the PANS method.

1 Introduction Various hybrid RANS/LES approaches have been proposed in recent years as alternative to a computationally more costly Large Eddy Simulation (LES) method. There are two main groups, namely zonal and seamless hybrid methods. The PANS approach is a novel seamless method by Girimaji et al. [5] and Girimaji [6], which changes seamlessly from RANS to DNS as the unresolved-to-total ratios of kinetic energy and dissipation are varied. Three variants of the PANS model are derived up to now, one based on the k-" formulation, the other based on the k-ω formulation (Lakshmipathy and Girimaji [7]) and the last one based on the ζ -f model recently proposed by Basara, Krajnovic and Girimaji [3]. Furthermore, Basara et al. [2] proposed a dynamic update of the unresolved-to-total ratio of kinetic energy f k B. Basara (B) AVL LIST GmbH, Advanced Simulation Technologies, 8020 Graz, Austria e-mail: [email protected]


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as a function of the mesh size and calculated length scales. The main target of such approach is to have an optimum turbulence model for any mesh used in calculations. The work presented in this chapter goes one step further and a dynamic value of f k is used as the criteria for an adaptive mesh generation. This means that calculations are performed on the original mesh and the parameter f k is calculated every time step from the length scale and the mesh size and therefore, it changes from point to point. In the work presented here, a dynamic f k serves as a criteria for an adaptive mesh refinement. All cells which have larger f k values than in advance prescribed value are refined. Consequently, f k is decreased and the PANS model captures smaller scales on the new mesh than on the original mesh without refinement.

2 Matematical Model The Partially-Averaged Navier – Stokes (PANS) equations are written in terms of partially averaged or filtered velocity and pressure fields, thus ∂τ (Vi , V j ) ∂Ui 1 ∂p ∂ 2 Ui ∂Ui + =− +ν + Uj ∂t ∂x j ∂x j ρ ∂ xi ∂x j∂x j

(1)

where the velocity field is decomposed into two components, the partially filtered component and the sub-filter component as Vi = Ui + u i .

(2)

The closure for the sub-filter stress can be obtained by using the Boussinesq approximation as 2 τ (Vi , V j ) = −2νu Si j + ku δi j 3

(3)

where, in the case of the PANS ζ -f model and following Basara et al. [3], the eddy viscosity of unresolved scales is equal to ν u = c μ ζu

ku εu

(4)

and the resolved stress tensor is given as 1 Si j = 2

∂U j ∂Ui + ∂x j ∂ xi

.

(5)

The model equations for the unresolved kinetic energy ku , the unresolved dissipation εu and the uresolved velocity scale ratio ζu are required to close the system of equations given above, thus

Enhancement of the Performance of the Partial-Averaged Navier – Stokes Method

νu ∂ku σku ∂ x j 2 εu νu ∂εu Dεu ∂ ∗ εu + = Cε1 Pu − C ε2 Dt ku ku ∂ x j σεu ∂ x j Dζu ζu ζu νu ∂ζu = f u − Pu + εu (1 − f k ) + Dt ku ku σζu ∂ x j Dku ∂ = Pu − εu + Dt ∂x j

335

(6)

(7) (8)

where 1 L ∇ fu − f u = Tu 2

2

P 2 c1 + c2 ζu − ε 3

(9)

where L is the length scale, and constants c1 and c2 are equal to 0.4 and 0.65 respectively, and Tu is the time scale defined by using unresolved kinetic energy. The model coefficients are (see also Girimaji [6]) ∗ Cε2 = C ε1 +

fk (Cε2 − C ε1 ) ; fε

σku,εu = σk,ε

f k2 fε

(10)

where the unresolved-to-total ratios of kinetic energy and dissipation are written respectively as fk =

ku , k

εu ε

fε =

(11)

It is obvious that the parameter f k given by Eq. (11) cannot be used for the input into the system of equations shown above as ku has to be solved. Note that a total kinetic energy is the summ of the unresolved and resolved kinetic energy. The parameter f k used in calculations, is based on the grid spacing, thus 1 fk = √ cμ

Δ Λ

2 3

(12)

1/3

where Δ is the grid cell size Δ = Δx Δ y Δz and Λ = k 3/2 /" is the turbulent length scale, while f ε was taken to be equal 1. The PANS asymptotic behavior goes smoothly from RANS to DNS with decreasing f k . Basara et al. [2] introduced f k given by Eq. (12) as a dynamic parameter in the computational procedure, changing at each point and at the end of every time step. This f k is then used as a fixed value at the same location during the next time step. Note that for f k = 1 and f " = 1, the equation for ζu will get its RANS form as well as all other equations. Note also that f ε = 1 implies εu = ε. The model is applied in conjunction with the hybrid wall treatment which combines the integration up to the wall with wall functions, see Basara [4].

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3 Numerical Procedure The model has been implemented into the commercial CFD code AVL FIRE [1]. The solution method is based on a fully conservative finite volume approach. The governing equations are integrated over the polyhedral control volumes. All dependent variables, such as momentum, pressure, density, turbulence kinetic energy, dissipation rate, and passive scalar are evaluated at the cell center. The cell-face based connectivity and interpolation practices for gradients and cell-face values are introduced to accommodate an arbitrary number of cell faces. A second-order midpoint rule is used for integral approximation and a second order linear approximation for any value at the cell-face. A diffusion term is incorporated into the surface integral source after employment of the special interpolation practice. The convection is solved by a variety of differencing schemes. The rate of change is discretized by using implicit schemes, namely Euler implicit scheme and three times level implicit scheme of second order accuracy which is used for all calculations presented in this chapter. The overall solution procedure is iterative and is based on the Semi-Implicit Method for Pressure-Linked Equations algorithm (SIMPLE). For the solution of a linear system of equations, a conjugate gradient type of solver and algebraic multigrid solver can be used. This solution procedure is straightforward usable for the grid adaptivity as refined cells and their neighborough cells are just treated as polyhedral cells, so no changes in the calculation algorithm. At the end, it is only about finding a connectivity between cells and applying a certain criteria to mark cells which should be refined. The grid adaptivity is the standard feature of AVL FIRE [1]. In this work, we used the parameter f k > 0.5 as the criteria for the mesh refinement.

4 Results A test case chosen for this work is the vortex shedding flow around a square cylinder at Re=21,400 (ERCOFTAC classic database). The dimensions of the domain are taken to be 20.5D × 14D × 4D, see Fig. 1 (left). Three Cartesian calculation grids are used: 30,000, 240,000 and 1,920,000 cells, denoted as MESH_01, MESH_02 and MESH_03 respectively. Some details of the coarsest mesh are shown in Fig. 1 (right). At the inlet of the solution domain, a uniform velocity profile is defined. At the outlet, a constant atmospheric pressure is prescribed. The periodic boundary is employed on the sides of the computational domain. An instantaneous unresolved-to-total ratio of kinetic energy f k computed as given by Eq. (12), and which is then used in Eqs. (6, 7, and 8), is shown in Fig. 2 (left). However, the parameter f k obtained from Eq. (11) by using predicted unresolved and total kinetic energy, and which is therefore contained in the solution, is shown in Fig. 2 (right). Note that these results are obtained at the same time step. Following procedure given in this chapter, the parameter f k which is used as the input parameter is always higher than the output f k , which means that the


337

y X

z

4D 14D FLOW 4.5D D

15D

Fig. 1 Computational domain (left) and the corsest mesh MESH_01 (right)

computational mesh supports input fk values, and the results should be improved when compared with RANS results. Note also that a distribution of the input and the output f k (Fig. 2) is very similar. The results obtained on three different meshes are shown in Fig. 3. It is clear that results are improved with the mesh refinement. Only results obtained on MESH_01 have a poor agreement with the measurements, for both, the time-averaged velocity and the time-averaged turbulent kinetic energy. It is important to note that calculations on the coarse mesh produce results on the level of the RANS accuracy as f k is equal or close to 1. The first adaptivity was performed on the MESH_01 and this mesh is refined from 30,000 to 245,000 cells. These results are denoted as an adaptive refinement 1 in Fig. 5. Comparing Figs. 2 and 5, it was concluded that a better strategy would be to start an adaptivity process from the MESH_02. Therefore, the MESH_02 was refined twice resulting with 408,000 cells and 696,700 cells. The final mesh is shown in Fig.4 (left, and denoted as an adaptive refinement 2) together with predicted instantaneous iso-surface of the second invariant of the velocity gradients (Fig. 4, right) as obtained on this mesh. Figures 5 and 3 clearly show that the accuracy of results is very similar between the refined

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 2 Unresolved-to-total ratio of kinetic energy computed by Eq. (12) (left) and by Eq. (11) (right)

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Fig. 3 Time-averaged velocity (left) and turbulent kinetic energy (right), along the centreline and behind the square cylinder, computed on three different meshes

Fig. 4 The result of the grid refinement applied on the MESH_02 (left) and predicted instantaneous iso-surface of the second invariant of the velocity gradients (right)

Fig. 5 Time-averaged velocity (left) and turbulent kinetic energy (right), along the centreline and behind the square cylinder, computed on the refined mesh of MESH_01 (Adaptive refinement 1) and on the refined mesh of MESH_02 (Adaptive refinement 2)


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mesh and the MESH_03, though almost double less cells were employed. However, it is not recommended to start this adaptive procedure on too coarse meshes as many refinement levels might be needed, which could lead to poor mesh quality and an accuracy of calculations can deteriorate.

5 Conclusions This chapter shows that applications with the hybrid RANS/LES approaches can be further optimized by using an appropriate scale-adaptive mesh generation. Computational costs can be adjusted to the hardware resources that are available at the given time by using the dynamic PANS ζ -f model in conjunction with here presented mesh adaptivity. Future work will include not just an adaptive mesh refinement but a mesh coarsening as well.

References 1. AVL AST, AVL FIRE Manual. AVL List GmbH, Graz (2009) 2. Basara, B., Krajnovic, S., Girimaji, S.: PANS vs. LES for computations of the flow around a 3D bluff body. In: Proceedings of ERCOFTAC 7th International Symposium On Engineering Turbulence Modelling and Measurements ETMM7, 2/3, 548-554. Lymassol, Cyprus (2008) 3. Basara, B., Krajnovic, S., Girimaji, S.: PANS methodology applied to elliptic relaxation-based eddy viscosity transport model, Proceedings of Turbulence and Interactions 2009 Conference. ISBN 3-642-14138-2, Springer, Berlin Heidelberg (2010) 4. Basara, B.: A non-linear eddy-viscosity model based on an elliptic relaxation approach, Fluid Dyn. Res. 41, 1–21 (2009) 5. Girimaji, S., Srinivasan, R., Jeong, E.: PANS Turbulence Models For Seamless Transition between RANS and LES: Fixed Point Analysis and Preliminary Results. ASME Paper FEDSM2003-45336, New York, USA (2003) 6. Girimaji, S.: Partially-Averaged Navier-Stokes Model for turbulence: A Reynolds- Averaged Navier-Stokes to Direct Numerical Simulation bridging method. J. Appl. Mech. 73, 413–421 (2006) 7. Lakshmipathy, S., Girimaji, S.: Partially-Averaged Navier-Stokes Method for Turbulent Flows: k- model Implementation, AIAA Paper 2006–119, Reston, USA (2006)

Sensitizing Second-Moment Closure Model to Turbulent Flow Unsteadiness Robert Maduta and Suad Jakirlić

Abstract Different scale-supplying equations formulated in the term-by-term manner at the Second-Moment Closure modeling level were a priori tested (the velocity and Reynolds stress data were taken from the available DNS database) in the flow in a plane channel in the Reynolds number range between Reτ = 395 and 2003 (DNS from Moser et al. (Phys. Fluids 11: 943–945, 1999) and Hoyas and Jimenez (Phys. Fluids 18: 011702, 2006)), the flow over a backward facing step (DNS: Le and Moin (J. Fluid Mech. 330: 349–374, 1997)) and the periodic flow over a 2-D hill utilizing the results of the highly resolved LES by Breuer (New Reference Data for the Hill Flow Test Case, http://www.hy.bv.tum.de/DFG-CNRS/, 2005). The starting basis of this activity are the model equations governing the total viscous dissipation rate ε and its homogeneous part εh = ε − 0.5ν∂ 2 k/(∂ x j ∂ x j ) proposed by Jakirlic and Hanjalic (J. Fluid Mech. 439: 139–166, 2003). The third equation governing the specific viscous dissipation rate ω = ε/k, i.e. ωh = εh /k has been directly derived from the corresponding εh -equation. Afterwards, the transport equation of the inverse turbulent time scale ωh is extended in line with the k −ω SST-SAS (Scale-Adaptive Simulation) model of Menter and Egorov (Notes on Numerical Fluid Mechanics, 2009) and applied to the afore-mentioned flow configurations in conjunction with the Jakirlic and Hanjalic Reynolds stress model equation (J. Fluid Mech. 439: 139–166, 2003).

1 Introduction It is well-known that the physics of the flows dominated by the organized, largescale coherent structures with a broader spectrum, as encountered in the flows involving separation, are beyond the reach of the conventional RANS (ReynoldsAveraged Navier – Stokes) method. There has been a substantial activity in developing the (seamless) hybrid LES/RANS methods (with an appropriately modified RANS model mimicking a subgrid-scale (SGS) model in the entire flow domain) S. Jakirlić (B) Institute of Fluid Mechanics and Aerodynamics/Center of Smart Interfaces, Technische Universität Darmstadt, D-64287 Darmstadt, Germany e-mail: [email protected]


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and novel Unsteady RANS methods (RANS model plays here the role of a subscale model). The relevant methods have been proposed by Menter end Egorov ([5]; SAS – Scale Adaptive Simulations), Girimaji ([3]; PANS – Partially-Averaged Navier Stokes) and Chaouat and Schiestel ([2]; PITM – Partially-Integrated Transport Model). The common feature of all these models is an appropriate modification of the destruction term (i.e. of its coefficient) in the scale-determining equation providing a dissipation rate level which suppresses the turbulence intensity towards the subgrid (i.e. subscale) level in the regions where large coherent structures with a broader spectrum dominate the flow, allowing in such a way evolution of structural features of the associated turbulence. Whereas an appropriate dissipation level enhancement in both PANS and PITM methods is achieved by reducing selectively (e.g. in the separated shear layer region) the destruction term in the model dissipation equation (i.e. its coefficient), an additional production term was introduced into the ω equation (ω ∝ ε/k – inverse turbulent time scale) in the SAS framework. This term is modelled in terms of the von Karman lenght scale comprising the second derivative of the velocity field (∇ 2 U), which is capable of capturing the vortex size variability, Menter and Egorov [5]. The work reported here aims at developing an instability sensitive SecondMoment Closure (SMC) model whose scale-supplying equation governing the ωh -variable is appropriately extended to behave as an SAS model.

2 Computational Model and Numerical Method The equation for the homogeneous part of the total viscous dissipation rate modelled in term-by-term manner by Jakirlic and Hanjalic [4] represents the starting point for the present development. The RSM-based ωh -equation following directly from the εh -equation by using well-known relationship Dωh /Dt = (Dεh /Dt)/k − εh (Dk/Dt)/k 2 (σω = σε = 1.1) reads: ωh νt ∂ωh ∂Ui 1 − (Cε,2 − 1)ωh2 ν+ − (Cε,1 − 1) u i u k 2 σω ∂ x k k ∂ xk 2 1 1 νt ∂ωh ∂k + + Pε,3 (1) ν+ k 2 σω ∂ xk ∂ xk k

Dωh ∂ = Dt ∂ xk

The “near-wall” term Pε,3 representing the gradient production (modelled by using the vorticity transport theory) comprising both the mean rate of strain and second derivative of the velocity field was also accounted for. The results of the a priori solving of the above equation (by taking the necessary input data from available DNS database) are shown in Fig. 1, left in the case of a fully-developed channel flow. The figure in the middle displays the homogeneous dissipation rate re-evaluated from ωh but also obtained by directly solving the εh -equation. The right figure depicts the reproduction of the correct profile shape of the total viscous dissipation rate. Similar results (agreement) are obtained by the a priori computations

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Fig. 1 Channel flow: results of a priori computations using different scale-supplying equations

of some other wall-bounded flow configurations (channel flow at a higher Reynolds number – Reτ = 2003, backward-facing step flow, flow over a 2-D hill; not shown here). The latter equation is appropriately extended through the introduction of the SAS term (Menter and Egorov [5]) into the ωh -equation:

Dωh,S AS Dωh = + PS AS ; PS AS = C R S M,1 max PS∗AS , 0 Dt Dt 1 (∇ωh )2 (∇k)2 L 2 ∗ 2 − 3k max C R S M,2 , PS AS = 1.755κ S L vk k2 ωh2 where L =

√

(2)

(3)

1/4 k/ Cμ ωh is the turbulent length scale, L vk = max(κ S/|∇ 2 U |;

C R S M,3 Δ) (Δ = (Δx Δ y Δz )1/3 ) represents the 3-D generalization of the classical boundary-layer definition of the 6 von Karman length scale and S the invariant of the mean strain tensor (S = 2Si j Si j ). It should be noted that the PS AS term introduced in the ωh -equation has almost identical form as the one being used in the eddy-viscosity-based k − ω SST-SAS model (Menter and Egorov [5]). However, the coefficients C R S M,1 = 0.008, C R S M,2 = 8 and exponent 1/2 (instead of 2), reducing substantially the intensity of the term, are introduced in order to adjust its use in the framework of a SMC model. Herewith, the RANS function

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of the present method is preserved within the near-wall region. The natural decay of the homogeneous isotropic turbulence, fully-developed channel flows in a range of Reynolds number (with underlying velocity field following the logarithmic law) and the non-equilibrium 2-D hill flow at Re H = 10600 have been interactively computed in the process of the coefficients calibration. The limiter C R S M,3 Δ in the L vk -formulation, originally introduced by Menter and Egorov [5], aims primarily at capturing correctly the turbulence spectra behaviour in the decay process of the homogeneous isotropic turbulence. However, this addition does not play an important role in two wall-bounded flow configurations considered. As the model validation in the homogeneous turbulence decay case is presently in progress, the value of the coefficient C R S M,3 is still not determined. The contours of the PS AS term depicted in Fig. 2 clearly show that it is active only in the region of the separated shear layer. In the reminder of the flow domain, especially in the near-wall regions, its effect vanishes. The magnitude of the original term (used in the k − ω SST – SAS framework) computed by using the flow field obtained by the present model is about 160 times larger (not shown here). All computations were performed using the code Open-FOAM [6], an open source Computational Fluid Dynamics toolbox, utilizing a cell-center-based finite volume method on an unstructured numerical grid and employing the solution procedure based on the implicit pressure algorithm with splitting of operators (PISO) for coupling between pressure and velocity fields. SIMPLE procedure was applied when computing the steady flows using the RANS-SMC model. The convective transport was discretized by the so-called “gamma scheme” (Jasak, 1996 PhD thesis, IC London), blending between 2nd order central differencing and 1st order upwind schemes with γC DS = 0.7 and γU DS = 0.3. For the time integration the 2nd order three point backward scheme was used.

Fig. 2 Magnitude of the PS AS term (Eq. 3) and instantaneous vorticity in the 2D hill flow field

3 Results and Discussion The model is illustrated by computing the periodic flow over a smoothly contoured 2-D hill at Re H = 10600 (reference LES by Breuer [1]) in the framework of the complete u i u j − ωh,S AS (denoted by SAS-RSM) model of turbulence (in conjunction with the Jakirlic and Hanjalic’s [4] Reynolds stress model equation). The solution domain L x × L y × L z = 9H × 3.03H × 4.5H (height H ) used for the


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Fig. 3 Periodic flow over a 2D hill, SAS-RSM and RANS-RSM results comparison: axial velocity (upper), turbulent kinetic energy (middle) and shear stress component (total stresses and their resolved fractions, lower) profile evolutions

computations was meshed by the grid 160 × 160 × 60 cells (denoted as “grid2”). The simulations were also performed on the substantially coarser grid comprising 80 × 100 × 30 cells (denoted as “grid1”). Maximum CFL number was about 0.3. It is interesting to report that no initial field fluctuations in these periodical flows were necessary. The fields obtained by the steady RANS computations served for the

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initialization of the computations with the present u i u j − ωh,S AS model. It should be pointed out that coupling the ωh transport equation (and not the one governing εh ) with the u i u j equation was presently preferred in order to “straightforwardly” introduce the PS AS term in its originally proposed form. Herewith, its transformation into the form usable in the εh equation was avoided. However, the εh variable was finally introduced into the Reynolds stress equation, after it has been recalculated from the ωh variable. The results, displayed in Fig. 2, right, document appropriate vortex structure reproduction - represented here by the instantaneous vorticity – being beyond the reach of any RANS model. The model capability to account for the large-scale structures and bulk unsteadiness led consequently to the increased magnitude of the turbulence kinetic energy (Fig. 3, middle; result typical of any RANS model is a significantly lower turbulence intensity in the separated shear layer; for comparison, the results of the u i u j − ωh model – denoted by RANS-RSM are also displayed), improved shape of the mean velocity profiles and correctly predicted reattachment length, Fig. 3, upper. The presented results reveal also a very small amount of modelled turbulence in the largest portion of the cross-section apart of the near-wall region, Fig. 3, lower.

4 Conclusions A near-wall second-moment closure model sensitized appropriately to account for the flow and turbulence instabilities was formulated in the present work. Accordingly, the scale-determining equation governing the inverse time scale ωh was extended in line with the SAS proposal due to Menter and Egorov [5]. After an appropriate “a priori” testing of the model in different wall-bounded flows, the feasibility of the formulation proposed solving the full set of the model equations proposed was checked by computing the flow over a 2-D hill. Promising results with respect to the structural characteristics of the instantaneous flow field, the mean velocity field and turbulence quantities illustrate the model potential in solving the flows separated from continuous curved surfaces exhibiting broader frequency range. It is interesting pointing out that no initially fluctuating velocity field was necessary for the present calculations. Further model validation in a range of complex wall-bounded flows is necessary. Acknowledgements The work has been funded by the EU project ATAAC (ACP8-GA-2009233710).

References 1. Breuer, M.: New Reference Data for the Hill Flow Test Case. http://www.hy.bv.tum.de/DFGCNRS/ (2005) 2. Chaouat, B., Schiestel, R.: A new partially integrated transport model for sub-grid-scale stresses and dissipation rate for turbulent developing flows. Phys. Fluids 17(065106), 1–19 (2005)


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3. Girimaji, S.S.: Partially-averaged Navier-stokes model for turbulence: A reynolds-averaged Navier-stokes to direct numerical simulation bridging method. J. Appl. Mech. 73, 413–421 (2006) 4. Jakirlic, S., Hanjalic, K.: A new approach to modelling near-wall turbulence energy and stress dissipation. J. Fluid Mech. 439, 139–166 (2003) 5. Menter, F.R., Egorov, Y.: Formulation of the Scale-Adaptive Simulation (SAS) Model during the DESIDER Project. In: W. Haase, M. Braza, A. Revell (eds.), Notes on Numerical Fluid Mechanics, vol. 103, pp. 51–62. Springer-Verlag Berlin Heidelberg (ISBN: 978-3-540-927723) (2009) 6. OpenFOAM – The Open Source CFD Toolbox: www.opencfd.co.uk/openfoam/

Part XIII

Turbulent Compressible/ Hypersonic Flows

High Order Scheme for Compressible Turbulent Flows Christelle Wervaecke, Héloïse Beaugendre, and Boniface Nkonga

Abstract The majority of fluid flows that are interesting from a practical point of view are turbulent flows. The problem is that turbulence is particularly difficult to model. Although simulation of turbulent flows has been the topic of important researches, it remains an open issue. We present a stabilized finite element method combine with the one equation Spalart–Allmaras model. In opposition to what is usually done, there is no splitting between the Spalart–Allmaras equation and the Navier–Stokes equations. The aim is to built high order scheme for compressible turbulents flows in order to control numerical dissipation and get accurate solutions.

1 Introduction On the one hand, all turbulent models include a turbulent viscosity as a parameter or a variable. On the other hand, numerical schemes always induce an artificial dissipation. This artificial dissipation is crucial to control, such as to be always lower to the viscosity obtained by subscale modeling of the turbulence. High order numerical approximations provide a framework where the constraint on the numerical dissipation can be achieved. Finite elements are suitable for the design of high order scheme with compact stencil that is efficient for parallel computing strategies by domain decomposition and messages passing. The main weakness of the classical finite element method (Galerkin) is its lack of stability for advection dominated flows. We consider in this work a compressible Navier – Stokes equations combined with the one equation Spalart-Allmaras turbulence model. These equations are solved in a coupling way. The numerical stability is achieved thanks to the Streamline Upwind Petrov – Galerkin (SUPG) formulation [2]. Within the framework of SUPG method, artificial viscosity is anisotropic and the principal component is aligned with streamlines. The aim is to put sufficient viscosity to get rid of instability and unphysical oscillations without damaging the C. Wervaecke (B) INRIA Bordeaux Sud-Ouest, Bordeaux, France e-mail: [email protected]


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accuracy of the method. The amount of artificial viscosity is controlled by a stabilization tensor τ . Since optimal way to choose τ is still unknown, several ways of computing τ are tested in this chapter. Besides SUPG method is also used in combination with a shock-parameter term which supplied additional stability near shock fronts [4].

2 Spalart-Allmaras Turbulence Model for Compressible Flows Let Ω ⊂ R d be the spatial domain with boundary Γ . The Navier – Stokes equations of compressible flows on Ω can be written as ∂Gi ∂U ∂Fi − = 0 + ∂t ∂ xi ∂ xi

(1)

where Fi and Gi are, respectively, the Euler and viscous flux vectors. Appropriate sets of boundary and initial conditions are set for Eq. (1). The turbulent kinematic viscosity νt = μt /ρ is, then, computed using Spalart-Allmaras (S-A) one-equation model ∂ρνt + ∇.(ρνt u) = M(νt )νt + P(νt )νt − D(νt )νt ∂t

(2)

where M(νt )νt represents the diffusion term, P(νt )νt the production source term and D(νt )νt the wall destruction source term. This model is an empirical equation that models production, transport, diffusion and destruction of the turbulent viscosity. See [3], for a complete description of the model. Navier – Stokes equations and Spalart-Allmaras equation are coupled in such a way that, the eddy viscosity is considered as an additional unknown of the system of equations (1).

3 SUPG Formulation The streamline upwind Petrov – Galerkin (SUPG) formulation is one of the most established stabilized formulations and is widely used in finite element flow computations. SUPG method introduces a certain amount of artificial viscosity in the streamline direction only. The aim is to prevent numerical instabilities without introducing excessive numerical dissipation.

3.1 Weak Formulation Consider a discretization of Ω into element subdomains Ωe , e = 1, 2, ..., n el , where n el is the number of elements. Given some suitable finite dimensional trial solution and test funtion spaces S h and V h , the SUPG formulation can be written as follows : find Uh ∈ S h such that ∀Wh ∈ V h

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Ω

Wh .

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n el ∂Wh ∂Gi ∂U h ∂Fi . Ai R (U˜ h ) dΩ = 0 − − S dΩ + τ + ∂t ∂ xi ∂ xi ∂ xi Ωe e (3)

where R(Uh ) is the residual of the differential equation: R(Uh ) =

∂Uh ∂Gi ∂Fi − − S + ∂t ∂ xi ∂ xi

(4)

The variational formulation (3) is built as a combination of the standard Galerkin integral form and a perturbation-like integral form depending on the residual. τ is the stabilization matrix, various options to compute this parameter were introduced in the literature. This will be discussed in the next section.

3.2 The Stabilization Parameter Stabilization parameter has been a subject of an extensive research over the last three decades. Nevertheless, the definitions of τ mostly rely on heuristic arguments and a general optimal way to choose τ is still unknown. Therefore, in this chapter, three ways of computing τ are tested. By analogy with formula for advection-diffusion problem, the parameter τ1 is defined by τ1 =

he Id ||u|| + c

(5)

Where h e is a measure of the local length scale, ||u|| is the flow velocity and the acoustic speed is defined as c. The second stabilization parameter proposed reads: n en −1 τ2 = |Ωe | A1 n ix + A2 n iy + A3 n iz

(6)

k=1

Ai =

∂ Fi ∂U

in Ωe

(7)

Indeed matrices Ai can be considered as a generalization of the scalar advection coefficient for the system of equation (1). Thus, these matrices can be employed in order to define the stabilization parameter. That is done following an expression coming from a Residual Distribution schemes, the N-scheme [1]. The third option has been proposed by T.E. Tezduyar and M. Senga [4]. τ3 =

n en

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Id

(8)

k=1

where j is a united vector defined as the tests functions.

∇ρ ||∇ρ|| ,

c is again the acoustic speed and φk are

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4 Numerical Results 4.1 Flat Plate Boundary Layer The first problem considered is a compressible high Reynolds number flow over a flat plate. Values of the CTTM are adopted, namely Cf = 0.00262 at Reθ = 104 and a Mach number of 0.24. See Spalart and Allmaras1 for more informations on the test case. The law of the wall and the log law are plotted with the computed solution. Theoretical and numerical results are in good agreement. The first mesh employed is viewed as fine. The first layer of nodes next to the wall is at y + = 0.5. To study the effect of mesh spacing, the closest point to the wall was allowed to vary. Figure (1) illustrates the effect of near wall refinement on the velocity profiles. For the sake of clarity, curves are shifted to the top on Fig. 1 while in fact, they overlay. There is very little variation in the results. Furthermore, no significant difference in the results has been observed for the three definitions of the stabilization parameter.

4.2 Backward-Facing Step The geometry is a 1.33 m long and 0.10 m tall rectangular inlet duct followed by a 0.0127 m rearward-facing step. The test was performed at a freestream velocity of 44.2 ms−1 and atmospheric total pressure and temperature. These conditions correspond to a freestream Mach number of 0.128, a boundary thickness of 1.5H, and 60

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a Reynolds number (based on momentum thikness) of 5,000 at a location 4 step heights upstream of the step. Figure 2a gives an overview of the flow field obtained, in wich a recirculation is formed downstream of the step. Several computations for this flow have been done. First, the SUPG method is used combined with the stabilization parameters τ1 , τ2 and τ3 . Solutions obtained with these different parameters are very similar. Then, a finite volume version of the same code is used to compare the finite volume solution to the SUPG one. Figure 2b represents the velocity profile at x = 2H . The fact underlined is that the SUPG method brings less numerical diffusion than a finite volume method.

4.3 Transonic Flow over a RAE2822 Airfoil The problem considered here is a flow over a RAE2822 airfoil. Fully turbulent flow was assumed, Reynolds number is 6.5 × 106 , Mach number is 0.734 and the angle of attack is 2.79◦ . Here, SUPG formulation is used in combination with a shockcapturing term [4] that provides additional stability near the shock front. Figure 3b

(a)

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Fig. 3 Velocity profiles for finite element and finite volume computations. (a) Pressure isolines, (b) pressure coefficient C p

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shows result for the coefficient of pressure C p . The mesh, used for this calculation, includes very stretched elements (aspect ratio up to 2 × 105 ). Due to this elongated cells, we still encounter some numerical instabilities. However C p obtained by the SUPG calculation is not so far from the experimental one. Some improvement in the handling of stretched element will certainly lead to better results.

5 Conclusion A stabilized finite element formulation applied to Navier – stokes equation combined with the Spalart – Allmaras model is proposed in this study. Numerical results show that, not only this method is able to reproduce good turbulent profiles in the case of a 2D flat plate, but also that it brings less numerical diffusion than a finite volume method. Even in the case of a almost incompressible flow, the numerical code is robust and gives good results. The choice of the stabilization parameter is yet a problem unsolved, but it seems that the second definition τ2 performed better with the test cases presented here. Acknowledgements Authors acknowledge Airbus France for its support through the DESGIVRE project.

References 1. Abgrall, R.: Essentially non-oscillatory: Residual distribution schemes for hyperbolic problems. J. Comput. Phys. 214, 773–808 (2006) 2. Hughes, T.J.R., Tezduyar, T.E.: Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput. Methods Appl. Mech. Eng. 45, 217–284 (1984) 3. Spalart, P.R., Allmaras, S.R.: A One-Equation Turbulence Model for Aerodynamic Flows. AIAA-92-0439 (1992) 4. Tezduyar, T.E., Senga, M.: Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Comput. Methods Appl. Mech. Eng. 195, 1621–1632 (2006)

Receptivity of Hypersonic Flow over Blunt-Noses to Freestream Disturbances Using Spectral Methods Kazem Hejranfar, Mehdi Najafi, and Vahid Esfahanian

Abstract The receptivity of supersonic/hypersonic flows over blunt noses to freestream disturbances is performed by means of spectral collocation methods. The unsteady flow computations are made through solving the full Navier-Stokes equations in 2D. A shock-fitting technique is used to compute unsteady shock motion and its interaction with freestream disturbances accurately in the receptivity study. The computational results for receptivity of a semi-cylinder at Mach 8 is presented and validated by comparison with available theoretical and numerical results. The study shows significant effects of the viscosity on the receptivity process.

1 Introduction One of the critical points in supersonic/hypersonic design specially for heating analysis of future vehicles is the laminar-turbulent transition prediction. Recently, a review on the effects of laminar-turbulent transitions upon high-speed entry vehicles performance revealed that understanding of transition and its origins, is a prelude [4]. The transition process, which is the nonlinear response of laminar boundary layer to freestream disturbances, conceptually has three important stages: receptivity, linear/nonlinear instability and breakdown to turbulence. Receptivity, which plays a key role in this process, stands for various mechanisms of transformation of external perturbations into internal ones. The stability and transition of supersonic and hypersonic boundary layers have been performed by many researchers. Formation of a shock wave in front of moving objects at supersonic/hypersonic regimes changes the receptivity of the boundary layer over them to freestream disturbances (Fig. 1). The bow-shock/disturbance interaction problem originates from studies of the receptivity in such flow regimes [1, 5]. While experimental and theoretical investigations are served as reference K. Hejranfar (B) Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran e-mail: [email protected]


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Fig. 1 A schematic of bow shock/freestream disturbances interaction in high-speed flow over a blunt body

points, up to now, they are very limited and hardly advisable for the study of different aspects of such problems where numerical tools are appropriate alternatives and are able to surmount. With the increase in computational power and development of high-order numerical methods, the study of the fundamental routes of transition to turbulence has become feasible. Such a simulation requires a good resolution of all relevant flow time and length scales, and therefore, highly accurate numerical methods are required. High-order compact methods have been used for studying the receptivity process [6, 7]. Although these schemes can provide accurate results for the receptivity problem, in practice they suffer from some difficulties like boundary closure problems. An alternative approach is to use spectral methods that do not have such deficiencies. In this work, the receptivity of viscous supersonic/hypersonic flow over blunt noses to freestream disturbances is studied by means of spectral collocation method. Both steady and unsteady flow computations are made through solving the full Navier – Stokes equations in 2D. The effects of bow-shock/freestream disturbance interactions on the receptivity process are accurately treated by taking the shock as a boundary, governed by the unsteady Rankine – Hugoniot relations. The computational results for receptivity of a semi-cylinder at Mach 8 is presented and validated by comparison with available numerical and theoretical results.

2 Problem Formulation The governing equations for this study are the two-dimensional Navier – Stokes equations, which can be written in the following conservation-law form in the Cartesian coordinates xi in terms of density, ρ, velocity, u i , pressure, p, temperature, T, ⎧ ⎫ ⎡ ⎧ ⎫ ⎤ ρ(u j − x˙ j ) ⎪ ⎪ 0 ρ ⎬ ⎨ ⎨ ⎬ ∂ ∂ 1 ⎦ ui + =⎣ ρu i (u j − x˙ j ) + pδi j ρ σx i ∂t ⎩ p ⎭ ∂ x j ⎪ γ ⎩ γ p + ρu k u k (u j − x˙ j ) ⎪ ⎭ γ −1 2 Pr ∇ · (k∇T ) + (γ − 1)Φ c (1)

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where ∂τij ∂u j ∂u i ∂u i 2 ∂u k σxi = . , Φ c = τij , τij = μ + − δij ∂x j ∂x j ∂x j ∂ xi 3 ∂ xk

(2)

In this chapter, the perfect-gas flow is considered, for which the equation of state is p = ρ RT where R is the gas constant. The ratio of specific heats is assumed to be constant (γ = 1.4). The viscosity coefficient μ is calculated by the Sutherland law. The extensive application of spectral collocation methods in the last decades for smooth partial differential equations showed superiority over other traditional methods. Globally accurate solutions and exponential decay of error with the increase in computational grid points are known to be a specific property of these methods [2]. However, the application of global spectral methods is limited to rather simple geometries and obtaining global smooth solutions, as the computational grid points are predetermined and the solution should be analytic over the whole computational region. Thus, the discontinuous feature of the shock wave in the present study is treated utilizing a shock-fitting technique. Details of the numerical algorithm can be found in [3].

3 Receptivity Simulation A weak monochromatic planar acoustic wave is introduced in the freestream to excite the instability waves in the shock layer. In this case, the perturbation amplitudes of nondimensional freestream flow variables satisfy the following dispersion relations: u = ε, ρ = εM∞ , p = γ εM∞ , u = 0, c∞ = (M∞ ± 1)√γ (3) ∞ ∞ 1∞ 2∞ where k∞ is the wavenumber related to the circular frequency ω by ω = k∞ c∞ . Also, ε is a small dimensionless number and εM∞ represents the relative amplitude of the wave. The plus and minus signs are used for the fast and slow acoustic waves respectively. The numerical simulation for the receptivity problem is carried out in two steps. First, a steady flow-field is computed by advancing the unsteady governing equations, with the time independent flow conditions. Second, a disturbance wave is imposed to the flow conditions by Eq. (3). After a periodic state is achieved, a wave decomposition procedure is performed on perturbed flow variables.

4 Results and Discussion A cylinder is placed in a hypersonic flow of M∞ = 8.03 with two Reynolds numbers of 2 × 103 and 2 × 105 based on nose radius. The (35 × 25) grid and steady state velocity vectors are plotted in Fig. 2 for Re∞ = 2 × 105 .

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Fig. 2 The grid, steady state velocity vectors and pressure contours for Re∞ = 2 × 105

After steady state reached, a monochromatic planar fast acoustic wave with ω = 101.76 is added to the freestream. The pressure fluctuation contours in the (x−t) plane along the stagnation line are depicted in Fig. 3. Two neighboring cells in time-direction have different shapes for the higher Reynolds number case and show a nonlinear behaviour, unlike the case for the lower one. This difference is most noticeable near the shock and becomes negligible near the body. The time history of instantaneous pressure just behind the bow-shock at the center-line is presented in Fig. 4. At the initial moments, there are no reflections from the undisturbed steady boundary layer and the figure shows a good agreement with the linear predictions of Morkovin [5] and the inviscid results of Chiu and Zhong [1] for the pressure amplitude. After it, there are two distinct regions, for both Reynolds numbers, that

Fig. 3 Instantaneous pressure fluctuation contours along the stagnation line for Re∞ = 2 × 103 and 2 × 105

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Fig. 4 Pressure variations just behind the bow-shock at the center-line for Re∞ = 2 × 103 and 2 × 105

the pressure amplitude and its time evolution differ considerably from the inviscid one and indicate the damping effect of viscosity on them. In Fig. 5 the amplitude of the pressure fluctuations along the stagnation line for Re∞ = 2 × 103 and 2 × 105 together with the inviscid computation of the developed code, Chiu and Zhong [1] and Morkovin [5] linear theory results are plotted. The wavelengths from numerical calculations are slightly larger than those of the linear theory. Numerical amplitudes of the pressure waves are also higher than those predicted by the linear analysis near the stagnation point. This discrepancy is caused by the simplified assumption of uniform steady flowfield behind the shock used in the linear analysis, which in reality is not uniform. Comparing the pressure

Fig. 5 Distribution of pressure fluctuation amplitudes along the center-line for Re∞ = 2 × 103 (left) and 2 × 105 (right)

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fluctuations amplitude for the viscous cases, the damping effect in the case with lower Reynolds number is obviously clear. As the Reynolds number increases, the viscous and inviscid results in this figure coincide with each other.

5 Conclusions A highly accurate spectral collocation method is developed for the receptivity study of viscous supersonic/hypersonic flows over blunt noses. The computational results for receptivity of a semi-cylinder at Mach 8 demonstrate the significant effects of viscosity on the receptivity process. The study demonstrates the capability and efficiency of the spectral methods in the receptivity study of supersonic/hypersonic flows over blunt noses. Acknowledgements The authors would like to thank Sharif University of Technology and University of Tehran for financial support of this study.

References 1. Chiu, C., Zhong, X.: Simulation of transient hypersonic flow using the ENO schemes. AIAA J. 34:4, 655–661 (1996) 2. Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods. Fundamentals in Single Domains. Springer, Berlin (2006) 3. Hejranfar, K., Esfahanian, V., Najafi, M.: On the outflow conditions for spectral solution of the viscous blunt-body problem. J. Comp. Phys. 228(11), 3936–3972 (2009) 4. Lin, T. C.: Influence of laminar boundary-layer transition on entry vehicle designs. J. Spac. Rock. 45(2), 165–175 (2008) 5. Morkovin, M. V.: Note on the assessment of flow disturbances at a blunt body traveling at supersonic speeds owing to flow disturbances in free stream. J. Appl. Mech. 27(6), 223–229 (2008) 6. Zhong, X.: High-order finite-difference schemes for numerical simulation of hypersonic boundary-layer transition. J. Comp. Phys. 144(2), 662–709 (1998) 7. Zhong, X., Tatineni, M.: High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition. J. Comp. Phys. 190(2), 419–458 (2003)

Part XIV

Bio-Fluid Mechanics

Flow Structures in Physiological Conduits A.M. Gambaruto and A. Sequeira

Abstract The study of topological structures in the case of a peripheral planar bypass graft and a cerebral aneurysm are presented. The Taylor series expansion of the velocity field to first order terms leads to a system of ODEs, the solution to which locally describes the motion of the flow. If the expansion is performed on the wall shear stress, critical points can be identified and the near-wall flow field parallel to the wall concisely described. Furthermore the expansion can be expressed in terms of relative motion and the near-wall convective transport normal and parallel to the wall can be accurately derived on the no-slip domain.

1 Introduction It is known that the haemodynamics in arteries is linked to disease formation such as atheroma and aneurysms. While the relationship between the flow field and disease are not fully understood, fluid mechanics parameters on and near the artery wall, such as wall shear stress and derived measures, are among the most commonly sought correlators to disease. Furthermore the non-planarity and tortuousity of vessels play a determining role in the arterial system, resulting in a strong influence of the local vessel topology on the flow field. In this work the flow structures for a peripheral bypass graft and a cerebral aneurysm, shown in Fig. 1, are studied for steady state simulations. In specific, the proposed methods are based on using existing theory of critical points [1, 2] as well as novel methods to look at the near-wall flow that is important in medical applications [3]. These will be used to observe the different topological structures in a detailed manner, and indicate a means to accurately formulate the near-wall transport from the no-slip domain. A.M. Gambaruto (B) Department of Mathematics and CEMAT, Instituto Superior Técnico, Lisbon, Portugal e-mail: [email protected]


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Fig. 1 (a) Cerebral vasculature and aneurysm (flow is from bottom to top) and (b) planar peripheral bypass graft (flow enters from top-left vessel and exits through top-right and bottom vessels)

2 Taylor Series Expansion of the Velocity Field Performing a Tarlor expansion of the velocity u i , i = 1, . . . , 3, in terms of the spatial coordinates xi we obtain u i = x˙i = Ai + Ai j x j + Ai jk x j xk + . . .

(1)

Let the coordinate system be such that the origin follows a fluid particle by translation and no rotation. In such a reference frame Ai = 0, while higher order terms are finite, and the origin is a critical point. By truncating the series to keep only the first-order term (the velocity gradient tensor), hence x˙ = (∇u) · x, or explicitly ⎛ ∂ x˙1 ⎞ ∂ x1 x˙1 ⎜ ∂ x˙ 2 ⎜ ⎝ x˙2 ⎠ = ⎝ ∂ x1 x˙3 ∂ x˙3 ⎛

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From this set of three first-order ODEs the eigenvalues (λi ) are obtained, which can be either three real values or one real and a complex-conjugate pair, the sum of which equals zero for incompressible fluids. The corresponding eigenvectors (ζi ) form planes to which the solution trajectories osculate, being either a node-saddlesaddle arrangement if the eigenvalues are real or a focus-stretching if a complexconjugate pair exist. An example of the solution planes is shown in Fig. 2 for a passive particle path in a vortical structure.

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The eigenvalues of ∇u satisfy the characteristic equation. λ3 + Pλ2 + Qλ + R = 0

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Furthermore, invariants of the rate-of-shear tensor and rate-of-rotation tensor are obtained from the above equations by setting in turn symmetric and anti-symmetric parts of the velocity gradient tensor to zero. The full set of invariants can be used to describe the local, instantaneous dynamics of the flow. A cross section example is shown in Fig. 3, where the in plane particle paths are used to explain the flow dynamics. It is clear that the eigenvalues and invariants can yield insight into studying and comparing the flow field locally in physiological situations. If the wall shear stress magnitude and direction are projected onto the piecewiselinear planar triangulated mesh that defines bounding geometry, then from a similar analysis the critical points of the traction force can be described. These can be either foci or saddle configurations. In doing so the near-wall flow parallel to the wall can be described elegantly as shown in Fig. 4.

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Fig. 3 Cross section of anastomosis with: (a) λ1 ; (b) |λ3 /λ2 | (to indicate the spiralling compactness for a focus configuration); (c) Q (the second invariant of the velocity gradient tensor); (d) velocity magnitude (m s−1 )

Fig. 4 Two views of the aneurysm surface with surface shear lines and locations of critial points of the wall shear stress, both foci and saddles. These are used to succinctly and clearly describe the near-wall fluid motion parallel to the wall

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Fig. 5 (a) Graft with spatial wall shear stress gradient sum and surface shear lines; (b) aneurysm with spatial wall shear stress gradient sum. Varying local intensity of flow movement to or from the wall is apparent and correlated approximately to the surface shear line divergence or convergence, respectively. A large motion of flow perpendicular to the wall is seen in the graft geometry while little is seen for the aneurysm.

3 Taylor Series Expansion of the Relative Position In a similar manner, but performing the expansion on the relative position instead, the near-wall convective transport is derived [3]. The leading terms relate the wall shear stress as the components parallel to the wall while the normal component is given by the sum of the spatial gradients of the wall shear stress, as shown in Fig. 5. It is apparent that the surface shear lines also shown in Fig. 5 indicate movement towards or away by their separating or coalescing, respectively (for constant wall shear stress). The wall shear stress, its spatial gradient sum [3] and its spatial gradient magnitude are commonly correlated to disease initiation and progression.

4 Conclusion and Future Work The Taylor expansion, while performed locally, acts as a perturbation analysis and is able to perceive the neighbouring flow field. As seen in [3] this permits measures from the flow field core to be extrapolated from the no-slip domain, allowing for a means to describe the flow field and hence the near-wall transport directly. In this work we have shown that the local dynamics for both the core flow and the near-wall flow (described using the traction force, hence wall shear stress) can be studied using the Talyor series expansion of the velocity or relative position. This leads to a local information of the flow field that can aid in discussing phenomena of human physiology in normal and diseased states. Future work includes using higher order terms to allow for a more detailed information of the flow field and a greater range in the locality of the analysis.

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References 1. Chong, M.S., Perry, A.E.: A general classification of three-dimensional flow fields. Phys. Fluids A. 2(5), 765–777 (1990) 2. Gambaruto, A.M., Moura, A., Sequeira, A.: Topological flow structures and stir mixing for steady flow in a peripheral bypass graft with uncertainty. Int. J. Num. Meth. Biomed. Eng. 26(7), 926–953 (2010) 3. Gambaruto, A.M., Doorly, D.J., Yamaguchi, T.: Wall shear stress and near-wall convective transport: Comparisons with vascular remodelling in a peripheral graft anastomosis. J. Comp. Phys. 229(14), 5339–5356 (2010)

Fluid Mechanics in Aortic Prostheses After a Bentall Procedure M.D. de Tullio, L. Afferrante, M. Napolitano, G. Pascazio, and R. Verzicco

Abstract The simultaneous replacement of a diseased aortic valve, aortic root and ascending aorta with a composite graft equipped with a prosthetic valve is a nowadays standard surgical approach, known as the Bentall procedure: the Valsalva sinuses of the aortic root are sacrificed and the coronary arteries are reconnected directly to the graft. In practice, two different composite-material prostheses are largely used by surgeons: a standard straight graft and the Valsalva graft with a bulged portion that better reproduces the aortic root anatomy. The aim of the present investigation is to study the effect of the graft geometry on the the flowfield as well as on the stress concentration at the level of coronary-root anastomoses during the cardiac cycle. An accurate three-dimensional numerical method, based on the immersed boundary technique, is proposed to study the flow inside moving and deformable geometries. Direct numerical simulations of the flow inside the two prostheses, equipped with a bileaflet mechanical valve with curved leaflets, under physiological pulsatile inflow conditions show that, when using the Valsalva graft, the stress level near the coronary-root anastomoses is about half that obtained using the standard straight graft.

1 Geometries and Materials Two prostheses can be used to perform the Bentall procedure [2]. One is the standard straight graft, with a constant orientation of the textile for the whole length. The other one is the Valsalva graft that exhibits three main portions, namely, the collar, the skirt and the body: while the collar and the body have the same properties of the straight tube prosthesis, the skirt is created by taking a section of graft and sewing it to the body with crimps in the vertical rather than in the horizontal direction. Figure 1 shows the geometrical model used in the numerical simulations. The model is discretized by triangular elements, as requested by the ray-tracing M.D. de Tullio (B) CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy e-mail: [email protected]


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Fig. 1 Model of the graft

technique used in the geometrical preprocessor [6]. The inflow tube, the valve housing and the collar are considered to be rigid, whereas the skirt, the body and the coronaries are discretized by 0.3 mm-thick linear elastic shell elements having large deflection capability. The same geometrical model is considered for both grafts, while the different behavior of the two ducts in the skirt region is taken into account by a different material model. The valve considered is a bileaflet 25 mm Bicarbon mechanical valve by Sorin Biomedica. The leaflets have a curved profile and exhibit a rotation range equal to 60◦ , with a fully open position of 10◦ with respect to the streamwise direction. As shown by Lee and Wilson [7], woven Dacron grafts can be characterized by orthotropic material properties, the stress-strain relationship being correctly modeled with a linear model in both the longitudinal and the circumferential directions. Here, the Dacron graft is considered as transversely isotropic, with the following elastic constants (z is the streamwise direction): Young’s moduli E x = E y = 12 MPa, E z = 1.2 MPa; Poisson’s ratios νx z = ν yz = 0.15, νx y = 0.1; Shear moduli G x z = G yz = 5.2 MPa, G x y = 0.55 MPa. It is noteworthy that for the Valsalva graft the orthotropic directions of the material are inverted at the skirt region, in order to take into account the different behavior of the prosthesis. A linear elastic material with Young’s modulus equal to 2 MPa is used for the two coronaries. The leaflets are made of pyrolytic carbon, with density of ρl = 2,000 kg m−3 , and are considered to be rigid.

2 Numerical Method and Simulation Details At every point of the time-dependent fluid domain, the Navier–Stokes equations for an incompressible viscous Newtonian fluid are solved, coupled with the ordinary differential equations governing the motions of the two leaflets. The fluid and the rigid leaflets are treated as elements of a single dynamical system, all governing equations being integrated simultaneously in the time-domain by a strong coupling scheme. Details on the basic fluid solver are given in Verzicco and Orlandi [9] and Fadlun et al [3], whereas the complete method and several numerical validations can be found in de Tullio et al [8]. Then, the Navier equations governing the dynamics

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of the deformable solid region are solved using the finite-element commercial softR [1]. Such a weak-coupling is employed in order to reduce the ware ANSYS computational cost of the overall procedure and to use optimized solvers for both the fluid and the elastic-body problems. In more details, at each time step, the following procedure is employed, where n and n + 1 indicate the old and new time levels, i = 1, 2 indicates the leaflets, subscripts f and s indicate fluid and structure quantities respectively: 1. iteratively solve the system of the fluid and leaflets’ equations using the positions and velocities of the structural nodes xns and uns as boundary conditions for the n+1 fluid domain, finding the fluid velocities and pressures, un+1 f , p f , as well as the leaflets angular positions and velocities, θ n+1 and θ˙ n+1 ; i

i

2. find the loads exerted on the structure by the fluid n+1 = n+1 (xns , un+1 f f f , n+1 n+1 n+1 p ,θ and θ˙ ); f

i

i

3. solve the structural equations with the computed loads, n+1 = n+1 s f , to obtain n+1 the new positions and velocities of the structural nodes, xs and un+1 s , to be used as boundary conditions for the successive time step. In this way, the solution of the structural solver is needed only once per time step. Stability is ensured by the very small time steps required by the flow solver to capture the time-history of the smallest turbulent scales. The coupling of the flow solver with the deformable structure has been validated considering a non-constrained pipe conveying a stationary flow: the present results, non reported here for brevity, agree well with the analytical predictions of the dimensionless pipe radius variation with dimensionless streamwise position at different Reynolds numbers, reported in [4].

3 Results and Discussion Typical physiological conditions for an adult male are considered. The cycle duration is set at 866 ms, corresponding to about 70 beats min−1 . The mean flow rate was adjusted to about 5 l min−1 with a peak flow rate of about 28 l min−1 . The blood kinematic viscosity and density are set to ν = 3.04 × 10−6 m2 s−1 and ρb = 1,060 kg m−3 , respectively. The peak Reynolds number, based upon the bulk velocity at the peak inflow, U = 0.95 m s−1 , and the inflow tube diameter is about 7,800. After a grid convergence study, a background cylindrical structured grid is used, having 217 × 165 × 250 nodes in the azimuthal, radial and axial directions respectively. For the fluid domain inflow section, the pressure and velocity profiles are imposed in order to mimic the physiological conditions produced by the heart in the left ventricle, as shown in Fig. 2a. Concerning the deformable structure, it is fully constrained at the lowest nodes (where the mechanical valve is fixed to the prosthesis), while a longitudinal displacement is applied to the highest nodes to simulate the longitudinal stresses reported in the natural aorta, according to the results of Han and Fung [5]. Here, the maximum displacement of about 5 mm is


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modulated in time by the pressure curve depicted in Fig. 2a. Five complete cardiac cycles are considered. The phase-averaged angular displacements of the two leaflets during the cardiac cycle are shown in Fig. 2b for both prostheses: very little sensitivity to the aortic root geometry is observed. The leaflet dynamics is greatly influenced by the pressure gradient through the valve, and this is evident during the opening phase, where the flow is accelerated. Small differences are noticeable during the closing phase, where the deceleration of the flow induces high turbulence levels, and therefore the forces on the leaflets are influenced by the fluctuations in time of the pressure and velocity fields. The out-of-plane vorticity contours in the symmetry plane are shown in Fig. 3, at three significant times. The typical configuration of the bileaflet

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Fig. 3 Out of plane vorticity in the symmetry plane for the Valsalva (top) and straight (bottom) grafts: (a, d) end opening of the leaflets; (b, e) flowrate peak; (c, f) start closure of the leaflets

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1.40E+06 valsalva straight tube

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valves, forming three jets, with strong shear layers shed from the valve housing and the tips of the leaflets is noticeable. At the peak flow rate, the shear layers become unstable and a strong small-scale turbulence production is observed in the wake and in the sinuses region. The Valsalva prosthesis exhibits an axisymmetric recirculation region, due to the sudden expansion flow past the valve, larger than that occurring for the case of the straight prosthesis and more similar to that occurring in the physiological situation. During the decelerating phase, the flow becomes turbulent with high production of small scale structures downstream the valve. The leaflets start closing as soon as the pressure starts decreasing and quickly reach a complete closure: the flow stops and viscosity dissipates the small scale structures until the beginning of the new cycle. Figure 4a provides the time histories of the Von Mises stresses near the coronary-root anastomoses for both the prostheses, while contour maps of the Von Mises stress distributions are plotted in Fig. 4b, at the minimum and maximum pressures registered during the cardiac cycle. The maps clearly show that the stress values, in the region corresponding to the sinuses, are smaller for the Valsalva prosthesis than for the straight one. In particular, the maximum stress for the Valsalva graft is about one half that for the straight one. The highest stress values are obtained for the Valsalva graft near the sinotubular junctions but they raise no concern because, in contrast with the hand-sewn coronary-root anastomoses, this zone is an all-Dacron component, reinforced by machine suturing during manufacturing, and thus it is not prone to rupture.

4 Conclusions The numerical tool developed by the authors provides a very detailed description of the unsteady flowfield inside two deformable Dacron prostheses carrying a bileaflet mechanical heart valve, under physiological pulsatile conditions. The code allows

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one to integrate the loads and thus to evaluate the stresses during the cardiac cycle. Results show that the dynamics of the leaflets is influenced only slightly by the prosthesis geometry, whereas near the coronary-root anastomoses the stress level for the Valsalva prosthesis having a bulged portion is about half that obtained using the standard straight tube. Therefore, the Valsalva graft could reduce complications like bleeding and pseudo-aneurysm formation, more likely to occur when using the straight one. This is the main and very interesting result of this work. Acknowledgements This research was funded by by MIUR and Politecnico di Bari under contract CofinLab 2001.

References R .: ANSYS Inc. http://www.ansys.com (2009) 1. ANSYS 2. Bentall, H., De Bono, A.: A technique for complete replacement of the ascending aorta. Thorax 23, 338–339 (1968) 3. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yosuf, J.: Combined immersedboundary finitedifference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161, 35 (2000) 4. Fung, Y.: Biodynamics: Circulation. Springer, New York, NY, Berlin, Heidelberg, Tokyo (1984) 5. Han, H., Fung, Y.: Longitudinal strain of canine and porcine aortas. J. Biomech. 28, 637–641 (1995) 6. Iaccarino, G., Verzicco, R.: Immersed boundary technique for turbulent flow simulations. Appl. Mech. Rev. 56, 331 (2003) 7. Lee, J., Wilson, G.: Anisotropic tensile viscoelastic properties of vascular graft materials tested at low strain rates. Biomaterials 7, 423–431 (1986) 8. de Tullio, M., Cristallo, A., Balaras, E., Verzicco, R.: Direct numerical simulation of the pulsatile flow through an aortic bileaflet mechanical heart valve. J. Fluid Mech. 622, 259–290 (2009) 9. Verzicco, R., Orlandi, P.: A finite difference scheme for threedimensional incompressible flows in cylindrical coordinates. J. Comput. Phys. 123, 402–413 (1996)

Optimisation of Stents for Cerebral Aneurysm C.J. Lee, S. Townsend, and K. Srinivas

Abstract Stents are used to effect a flow diversion in an aneurysm to reduce the risk of its rupture. Following the technique of exploration of design space, the present work attempts to optimize the design of stents. In this study both two-dimensional simplification of stents and the three-dimensional ones are considered. Optimization determines the most effective arrangement of struts and gaps within the stents. Velocity and vorticity reduction within the aneurysm form the objective functions. Optimisation is performed by considering random design of stents and arriving at what are called Non-Dominated solutions. The designer identifies the one that suits the requirements best. It is shown that for best flow diversion, a strut at the proximal end is a necessity.

1 Introduction Aneurysms are caused by pathological dilation of the arterial wall. They can grow in size, which can cause internal hemorrhage leading to stroke or death [4]. To avoid rupture, stents are used to cause a flow diversion, i.e., reduce flow activity within. A number of experimental and computational studies of the flow in a stented aneurysm are available in the literature. In particular, detailed studies on a two-dimensional simplified model of stented aneurysms carried out by Hirabayahsi et al. [1] show that the position of gaps and struts, and their size has a considerable effect on the flow diversion caused. One of our objectives in the present research is to take the next step, and optimize the design of the stents for an effective flow diversion. This is similar to the work reported by Srinivas et al. [5], which considered optimization of stents for coronary arteries. In this work we present two-dimensional results for aneurysms and some preliminary results for three-dimensional case.

C.J. Lee (B) School of Aerospace, Mechanical and Mechatronic Engineering, University of Sydney, Sydney, NSW 2006, Australia e-mail: [email protected]


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2 Methodology Optimisation follows the Exploration of Design Space [2, 3, 5] approach. To start with a large number of samples called individuals are selected within the range of design variables using the Latin Hypercube technique [3]. Objective functions are computed for each of these samples. An optimization procedure based on Kriging [3] is then carried out. What results will be a set of optimum or non-dominated solutions. The designer selects the one that suits best for a given situation.

2.1 Stents Considered Both two- and three-dimensional aneurysm geometries are considered in this study, as shown in Fig. 1. For a two-dimensional simplification, the stent placed at the neck of the aneurysm cavity appears as an arrangement of struts and gaps, whereas for a three-dimensional model the stent is created using Pro Engineer to be a close resemblance to a real stent. Many different cases of two-dimensional stents were considered in the study; only a typical one is presented here. This consists of five struts of uniform width, each being 0.4 mm with varying gap sizes. The porosity in this case is 80% while the thickness of each strut is 0.1 mm. The other case presented is a three-dimensional one. At the time of writing this chapter, only one three-dimensional case has been completed. Similar to the two-dimensional case there are five struts of uniform width and thickness, 0.4 and 0.1 mm, respectively. Only the gap sizes are varied, with the mid-cross section porosity fixed to 80%. A typical three-dimensional stent model is shown in Fig. 2. The results for this case will be presented as Case 2 in the Results section.

2.2 Objective Functions As mentioned before, the aim of stent design is to reduce the flow velocity within the cavity. In addition, vorticity is a good indicator of viscous activity. Hence, the following objective functions are chosen.

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Fig. 2 A typical three-dimensional stent model created using Pro-Engineer S) 1. Velocity Reduction in the cavity, expressed as a percentage, ΔV = (V NVSN−V × S 100, where V N S , and V S , are the area weighted average of velocity magnitude within the cavity without the stent and in presence of the stent, respectively. 2. Vorticity = 5 Reduction in the cavity, expressed as a percentage, Δω 5

ω N S − d ωS 5 d ωN S

× 100, where ω is vorticity, NS and S, represent within the cavity without the stent and in the presence of the stent, respectively, and d denotes the interior of the cavity. d

Both of the above objective functions are to be maximized.

2.3 Computational Details All aneurysm and stent models were modeled and meshed using GAMBIT 2.3, with approximately 100,000 grid nodes within the region and approximately 350 points specified on the cavity surface, using triangular meshing scheme. The fluid considered is blood with a density of 1, 060 kgm−3 , a viscosity of 0.0035 kg/m·s, and the flow velocity at the entry is 0.3 ms−1 . The Reynolds Number based on entry conditions is 363. The flow in the vessel is considered as steady and Newtonian. The governing equations are those for a steady, two-dimensional laminar flow. The boundary conditions applied are the no flow conditions on all solid boundaries, constant velocity condition at the flow entry and outflow conditions at the exit. All computations were performed using FLUENT 6.3, on personal computers available at the University of Sydney, with the CPU time of between 2 min and 1 h depending upon the case considered. Computations for three-dimensional models required CPU times around five hours. A number of grid dependence studies were performed for the no-stent case and for a few of the stented cases. The solution was deemed to have converged when the residual for each of the equations was reduced to a level below 10−6 . The next step is to generate what are known as non-dominated solutions using Kriging explained in Srinivas et al. [5]. From these a compromise solution is identified and forms the optimum solution or stent.

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3 Results and Discussion Figure 3 shows the velocity vectors and vorticity contours for the no-stent case. Case 1 Forty random samples were considered for this case. The objective functions are plotted in Fig. 4, which also shows the non-dominated solution for this case and consists of as many as 30 solutions lying close to each other. Figure 5 shows the results for the best, the worst and the compromise stent identified in the non-dominated solutions for this case. The results for this case clearly indicate that for a good flow diversion, one should place struts at the proximal end of the neck.

Fig. 3 Velocity vectors (left) and vorticity contours (right) for no-stent case

Fig. 4 Non-dominated solutions for case 1

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Fig. 5 Velocity vectors (top) and vorticity contours and strut arrangements for case 1, (a) best stent and (b) worst stent and (c) compromise stent

Case 2 Figure 6 shows the results for the best and the worst stent for this case. The threedimensional results are in agreement with the results from two-dimensional case that for a good flow diversion, one should place struts at the proximal end of the neck. Velocity reduction in the dome ΔV had a minimum of 20.55% while its maximum

Fig. 6 Velocity (top) and vorticity contours and strut arrangements for case 2, (a) no-stent, (b) best stent and (c) worst stent

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was 36.66%, reduction in vorticity Δω varied between 8.9 and 23.38%. Rigorous optimization is being carried out now.

4 Conclusion Two-dimensional and three dimensional stents have been optimized for an effective flow diversion for an aneurysm. The stent consists of five struts of uniform width and thickness; the design variable being the gap sizes. It is revealed that a strut at the proximal end of the aneurysm is essential for a good flow diversion.

References 1. Hirabayashi, M., Ohta, M., Barath, K., Rufenacht, D.A., Chopard, B.: Numerical analysis of the flow pattern in stented aneurysms and its relation to velocity reduction and stent efficiency. Math. Comput. Simulat. 72, 128–133 (2006) 2. Jeong, S., Obayahsi, S.: Multi-objective optimization using Kriging model and data mining. KSAS Int. J. 7, 1–12 (2006) 3. Myers, R.H., Montgomery, D.C.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments, pp. 1–84. Wiley, New York, NY (1995) 4. Sforza, D.M., Putman, C.M., Cebral, J.R.: Hemodynamics of cerebral aneurysms. Ann. Rev. Fluid Mech. 41, 91–107 (2009) 5. Srinivas, K., Ohta, M., Nakayama, T., Obayashi, S., Yamaguchi, T.: Studies on design optimization of coronary stents. J. Med. Dev. 2, 011004-1–011004-7 (2008)

Micron Particle Deposition in the Nasal Cavity Using the v2 -f Model Kiao Inthavong, Jiyuan Tu, and Christian Heschl

Abstract From a health perspective inhaled particles can lead to many respiratory ailments. In terms of modelling, the introduction of particles involves a secondary phase (usually solid or liquid) to be present within the primary phase (usually gas or liquid). The influence of the fluid flow regime, whether it is laminar or turbulent plays a significant role on micron particle dispersion. RANS (Reynolds Averaged Navier-Stokes) based turbulence models provide simpler and quicker modelling over the more computationally expensive Large Eddy Simulations. However this comes at an expense in that the RANS models fails to resolve the near wall turbulence fluctuating quantities due to the turbulent isotropic assumption. This error further propagates to the Lagrangian particle dispersion. Using the v2 -f the normal to the wall turbulent fluctuation, can be solved and used on the particle dispersion model directly in order to overcome the isotropic properties of RANS turbulence models. This technique is first validated against experimental pipe flow for a 90o -bend and then applied to particle dispersion in a human nasal cavity using Ansys-Fluent. The results arising from the nasal cavity application will increase the understanding of particle deposition in the respiratory airway. Greater knowledge of particle dynamics may lead to safer guidelines in the context of exposure limits to toxic and polluted air.

1 Introduction Studies of gas-particle flows in the human nasal cavity have generated a lot of interest recently as computational modelling offers a complementary alternative to experimental methods. In particular the inhalability of particles has been studied which showed aspiration efficiencies of 60–80% [4] and 50–95% [3] for micron particles between 1–40 μm. In terms of modelling, the introduction of particles involves a secondary phase (usually solid or liquid) to be present in conjunction with the primary phase (usually gas or liquid), leading into the field of multiphase flows. The K. Inthavong (B) RMIT University, Bundoora, Australia e-mail: [email protected]


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dispersed phase can be modelled under two different approaches, i.e. Lagrangian or Eulerian. Both approaches have their own advantages in computational modelling, however this chapter is limited to the Lagrangian approach only. Micron particles are dominated by its inertial property which lead to inertial impaction upon sudden changes in the airflow streamlines. When the flow field is turbulent, turbulent dispersion of the particles has to be addressed. Reynolds Averaged Navier Stokes (RANS) based turbulence models are often used to resolve the flow field, as it provides simpler and quicker modelling over the more computationally intensive Large Eddy Simulations. However this comes at an expense in that the RANS models fails to resolve the turbulence dissipation and anisotropy in the near wall regions (i.e. fixed stationary surfaces or boundaries). The turbulence fluctuating quantities are overpredicted by RANS models and this error propagates to the particle dispersion. In this chapter, the v2 -f and the k-ε turbulence model are used to solve the fluid flow, while a dispersed phase model (Lagrangian reference) is used to track the individual particles. Turbulent dispersion of particles is modelled by the so-called Discrete Random Walk found in Ansys Fluent. The turbulent particle tracking scheme is evaluated and the requirements for the models to account for the anisotropic flow behaviour in the near wall region is discussed. Successful modelling of micron particles will allow more flexibility in simulations of gas-particle flows for inhalation toxicology, and drug delivery studies through the human respiratory system.

2 Model Description The 90o curvatures in the nasal cavity are located just after the nostril entrance, and at the nasopharynx region (Fig. 1). These two bends act as a naturally occurring filter system that traps high inertial particles as they travel through the airway. A simpler test case for evaluation of the CFD modelling is to use a 90o -bend pipe based on experimental [6] and Large Eddy Simulation [2] data (Fig. 1). The computational pipe (0.6 million cells) has a diameter of 0.02, radius of curvature Rb = 0.056, a

Fig. 1 (a) Lateral view of the nasal cavity model showing the left cavity side. Two 90o curvature bends are present; one at the nostril inlet and the other at the nasopharynx region. Three crosssectional slices are made at the anterior, middle and posterior regions labelled as A, B, and C respectively. (b) Lateral view of the 90o bend pipe

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curvature ratio of Ro = 5.6, Re = 10,000 and De = 4,225. The nasal cavity (3.5 million cells) is subjected to a Re = 2,498 at the outlet which corresponds to a flow rate of 20 L/min. The realizable k-ε turbulence model with enhanced wall function is applied through Ansys–Fluent and its model equations are provided in Ansys–Fluent (Ansys 2009). The v2 -f model was implemented in Fluent via user-defined scalar interface (UDS). The segregated solver in Fluent is used to solve the additional transport equations for v2 and f, and therefore the code friendly v2 -f version by Lien and Kalitzin [5] is applied to improve convergence. For the Discrete Phase Model (DPM) the Lagrangian approach is used. The normal (perpendicular to the wall) fluctuation component, (v ) in the near wall region (y+ ≈ 0–30) is significantly damped in comparison to the corresponding fluctuating components, u and w . With the v2 -f model, the turbulent fluctuation component perpendicular to the wall is resolved which can then be applied to the DRW model in the near wall region. The modified DRW model applied to regions where y + < 30 is then reformu 0.5 lated as v = ζ v 2 . The number of droplets tracked was checked for statistical independence since the turbulent dispersion is modelled based on a stochastic process. Independence was achieved for 40,000 droplets since an increase of droplets to 60,000 droplets yielded a difference of 0.1% in the inhalation efficiency. To achieve the uniform droplet concentration assumption, droplets were released at the same velocity as the freestream.

3 Results and Discussion Velocity contours at the 45o and 90o deflection are shown in Fig. 2. It can be seen that the high velocity region moves from the inner wall to the outer wall (from the 45o deflection to the 90o deflection). A larger region of slower velocity is found at the inner wall region as the secondary flow effects progress with the flow moving through the bend from 45o to 90o . In addition the streamlines highlight the movement of the fluid from the core towards the outer pipe wall with two resultant vortices near the inner wall region. The secondary flow features are captured well with the LES model by [2]. The v2 -f model was also able to capture some of the secondary flow effects although at a reduced level. The region of slow moving fluid is much smaller than that of the LES data especially at the 45o deflection. Even worse performing is the k-ε model which does not capture the slow moving region at all at the 45o deflection. Furthermore, the streamlines in the core flow, moving towards the outer wall are distorted at the 90o deflection which is not reproduced by the two RANS turbulence models. The normal fluctuating velocity component (v’) taken at the pipe bend entrance is chosen for comparison with DNS data because this is the component that is overpredicted by the DRW models when a RANS-based turbulence model is applied. In the DRW √ model, the particle takes the fluctuating velocity component as v = u = w = 2k/3. Figure 3 shows that when this occurs the v’ component is overpredicted in both the v2 -f and the k-ε model. Because the v2 -f model provides directly

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Fig. 2 and k-ε simulation results compared with LES data by Berrouk and Laurence [2]. The angled deflections indicate the pipe curvature location. The inside wall of the pipe is on the left side and the outer wall of the pipe is on the right indicated by I and O respectively

the v2 profile near the wall, the v2 can be defined directly into the DRW model. Its profile is shown in Fig. 3 by the dashed line and denoted as v2-f(v2) which shows a better improvement in the near wall region. Therefore the proposed modification of the DRW model is expected to improve the turbulent particle dispersion. The predicted deposition of 1–30 μm particles in a 90o bend pipe compared with the experimental data of Pui et al. [6] is shown in Fig. 3b, where the particle Stokes number is based on the pipe inlet conditions. The deposition efficiency for the DRW model taking the default isotropic fluctuating component from k (turbulent kinetic b 1.5

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energy) shows an overprediction in the deposition for St < 0.1 (square symbols in Fig. 3b). The overprediction becomes greater, away from the experimental data as the St decreases further. When the DRW model is modified and the normal fluctuating component takes on the v2 profile, the deposition efficiency for St < 0.1 is improved. Particle tracking analysis using the DRW-mod-v2 -f model is performed and the coordinates of the particles as they move through specified slice planes in the geometry are recorded. This can give an indication of how the particles are moving through the nasal cavity, and also how many are passing through. Three crosssectional slices, A, B, and C created (from Fig. 1), are viewed from the front of the nose which is shown in Fig. 4. Slice A exhibits the highest maximum velocity, with streamlines directed toward the inner nasal septum wall. At this slice, only 77% of 15 μm particles pass through, meaning that ≈ 23% has deposited already in the anterior nasal cavity (nasal vestibule) region. The particles are concentrated close to the ceiling of the passageways with high velocities. The streamlines in Slice A tell us that the secondary flow will push the particles both upwards and downwards.

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This results in the 1 μm particle dispersing more evenly through Slice B whereas the 15 μm particles remain close to the top of the slice because of its high inertial property. All particles are mainly passing through the inner side of the passageway (the nasal septum wall side). For the 15 μm particles there is a large drop in the number of particles passing through from Slice A to Slice B ( 61%), meaning that this is the main section of deposition for the particles. Interestingly this is also the main deposition region for the 1 μm particles with a percentage deposition of 3.3%. At Slice C, the airflow between the left and right nasal cavity chambers have merged. Here we see complex secondary flow patterns, exhibiting two vortices and two peak axial velocity regions each of which are almost symmetrical to each other. The streamlines from the left and right sides converge in the middle of the slice and are directed towards the inner curvature wall side of the slice. Nearly all the particles moving from Slice B to Slice C have passed through the passageway which has now expanded in cross-sectional area. The main cluster of particles for the 15 μm particles is still found in the superior regions of the slice, which is now the outer curvature wall side. These flow patterns provide a predictive tool as to where the particles may travel. The particle tracking model can be used to determine the localised regions of high particle deposition.

4 Conclusion The Discrete Random Walk (DRW) model used in Ansys-Fluent was used to simulate dispersed particles through a wall-bounded geometry such as the human nasal cavity. A breathing rate of 20 L/min was used and a RANS based turbulence models in the form of the k-ε and the v2 -f √ model was applied. It was shown that by applying the small modification of v = v 2 directly the DRW can take on a more realistic turbulent dispersion in the near wall region. A 90o pipe bend was used to validate the model which showed an improvements to the particle deposition. The nasal cavity was then used as an application which also showed that the modified DRW model improved the deposition efficiency for smaller inertial particles.

References 1. Abe, H., Kawamura, H., Matsuo, Y.: Direct numerical simulation of a fully developed turbulent channel flow with respect to the Reynolds number dependence. J. Fluids Eng. 123, 382–393 (2001) 2. Berrouk, A.S., Laurence, D.: Stochastic modelling of aerosol deposition for les of 90◦ bend turbulent flow. Int. J. Heat Fluid Flow 29(4), 1010–1028 (2008) 3. Kennedy, N.J., Hinds, W.C.: Inhalability of large solid particles. J. Aerosol Sci. 33, 237–255 (2002) 4. King Se, C.M., Inthavong, K., Tu, J.: Inhalability of micron particles through the nose and mouth. Inhal. Toxicol. 22(4), 287–300 (2010) 5. Lien, F., Kalitzin, G.: Computations of transonic flow with the v2-f turbulence model. Int. J. Heat Fluid Flow 22, 53–56 (2001) 6. Pui, D.Y.H., Romay-Novas, F., Liu, B.Y.H.: Experimental study of particle deposition in bends of circular cross section. Aerosol Sci. Technol. 7(3), 301–315 (1987)

Part XV

Meshing Technology

Adjoint-Based Adaptive Meshing in a Shape Trade Study for Rocket Ascent Marshall R. Gusman, Jeffrey A. Housman, and Cetin C. Kiris

Abstract This paper investigates the drag performance of six axisymmetric shroud shapes over the ascent trajectory of the Saturn V launch vehicle. An adjoint-based adaptive meshing algorithm is used to generate Cartesian meshes with refinement in regions of the flow that have a critical influence on drag prediction. In combination with an efficient parallel flow solver, this approach provides accurate predictions of aerodynamic performance at minimal computational cost. A metric known as ‘drag loss’ is integrated over the flight trajectory to determine the best shroud shape for the application. These methods are shown to be an effective aid to the aerodynamic design of new launch vehicles.

1 Introduction NASA has renewed interest in traditional rocket designs such as the Saturn V, which successfully launched many orbital and lunar missions during the Apollo program of the 1960s and 1970s. The “vertical stack” design places the payload at the top of the rocket where a significant fraction of total drag force is generated. The current paper seeks to improve the aerodynamic performance of a “vertical stack” launch vehicle by identifying the best of six shroud shapes for a Saturn V geometry and ascent trajectory. This study is performed with the NASA-developed software Cart3D, a parallel CFD package that includes an embedded boundary automatic Cartesian mesh generator and an inviscid flow solver, see Aftosmis et al. [1] as well as an adjoint-based mesh refinement capability, see Nemec and Aftosmis [3], and Nemec et al. [4]. Solutions are marched in time to steady-state using a five-stage Runge-Kutta scheme with local time stepping and a multigrid W-cycle. The inviscid flow assumption is valid provided boundary layer effects are negligible over the region of interest. The combination of large Reynolds numbers and relatively short shroud lengths results M.R. Gusman (B) ELORET Corp., Sunnyvale, CA 94086, USA e-mail: [email protected] This work was performed under NASA contract NNA06BC19C to ELORET Corp. A. Kuzmin (ed.), Computational Fluid Dynamics 2010, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-17884-9_49,

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in minimal boundary layer development and negligible skin friction effects. The software’s parallel efficiency and solution-adaptive meshing provides an excellent utility for shape trade analyses in the early stages of rocket design.

2 Adaptive Meshing Adaptive mesh refinement is an intelligent alternative to a global refinement approach because it requires fewer resources to achieve the same level of solution accuracy. Conversely, better accuracy can be achieved with the same resources by adaptively refining the mesh in only the important regions. The adjoint-based adaptive mesh refinement method involves choosing a functional of interest, J (Q), which is a scalar quantity that depends on the flow solution, Q. In this study, the functional is the drag force on the shroud. The accuracy of the flow solution and functional approximation are highly dependent on the discretization error associated with the numerical scheme and computational domain. The discretization error can be reduced by globally refining the mesh, but we soon discover that global refinement is inefficient and unaffordable. A more cost-effective approach is to restrict refinement to the most critical regions of the flow domain.

2.1 Adjoint Method The critical regions are identified by solving the adjoint (or dual) problem. This solution can be used to produce error estimates representing each cell’s contribution to the global error in the functional. A user-specified error tolerance criterion is then used to identify which cells should be refined to improve accuracy. After obtaining the newly refined mesh, the flow solution is recomputed and the adaptation procedure is repeated until the functional error tolerance is met. The following procedure identifies the critical regions and refines the mesh to improve the functional estimate. Functional Approximation: From a given solution, the functional on a uniformly refined mesh can be approximated with a Taylor series expansion about a solution Q hH , which is interpolated from the current mesh (H ) to the uniformly refined mesh (h).

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Q h − Q hH . J (Q h ) ≈ J Q hH + ∂ Qh

(1)

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Setting R(Q h ) to zero and rearraging Eq. (2) gives the following expression

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Adaptation Criteria: The functional approximation described above contains a correction term and a term representing the remaining error. Since the correction term is computable, it can be used to correct the functional on the given mesh. The remaining error term cannot be affordably computed and must be driven to zero by refining the cells which contain the largest remaining local errors. Since the remaining error is not known, the local error estimates are approximated with trilinear and triquadratic interpolations of ψ H onto the embedded fine mesh, denoted ψT L and ψT Q respectively. This leads to the following local error estimate for each cell,

T (7) e = ψT Q − ψT L R Q hH . Next, a user-defined tolerance εT O L is used to define a threshold representing the required accuracy of the computed functional. Since the remaining global error is simply the sum of the remaining local errors over all cells, then ε L OC AL = εT O L /N where N is the total number of cells in the current mesh. Thus a cell is flagged for refinement if e > ε L OC AL . Solve and Repeat: A newly refined mesh is generated based on the flags from the adaptation criteria. The flow solution is recomputed on the fine mesh as before, and the procedure is repeated until the estimated remaining global error is less than εT O L .

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2.2 Adapted Meshes The adapted meshes show that refinement is limited to the regions affecting the functional of interest (drag on the shroud). Figure 1 shows a cutting plane of the adapted mesh at successive levels of refinement for the baseline shape in supersonic flow. Mesh points cluster where the largest errors are estimated; at the leading shock structure and near the body, while regions downstream of the shroud are ignored. Localized refinement keeps the computational cost to a minimum while capturing a high level of accuracy. A slice of the mesh for subsonic flow in Fig. 2a shows refinement upstream and downstream of the shroud. The adjoint-based adaptation detects flow influence from both directions and refines appropriately. A slice of the transonic mesh (b) reveals refinement upstream of the shroud and an abrupt halt to refinement after the shroud. This is caused by a supersonic bubble that exists just downstream of the shroud and prevents information and error from traveling

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upstream and affecting the shroud. With supersonic flow, downstream refinement is unnecessary. When the freestream flow is supersonic, even less refinement is necessary. The mesh is only refined in regions of high error (like shocks) that influence the functional, and all other regions are left coarse. The critical diamond-shaped region of influence and dependence is apparent in Fig. 1f. Upstream of the bow-shock the flow is uniform and there is no error in the solution, and downstream of the shroud there is zero functional sensitivity to errors in the solution. Refinement only occurs where the product of residual- and adjoint-sensitivity is high. A brief verification of the adaptive meshing procedure is presented for the baseline shape at Mach 1.7. Figure 3a shows the drag coefficient (CD) as a function of the total number of cells as the mesh is refined. Initially, the 22,000-cell mesh under-predicts CD by approximately 7% in comparison to the drag estimate on the final 3,800,000-cell mesh. The corrected functional is also shown, which is calculated using the computable correction term from Eq. (6). On a given mesh, the corrected functional is an estimate of the functional value that would be obtained on a mesh uniformly refined by one extra level. Both values converge towards C D ≈ 0.805. Progressive improvements to the functional accuracy are observed in Fig. 3b, where the functional error estimate is plotted versus mesh size. The log of error falls linearly with the log of the number of cells. Extrapolating this linear behavior to the final mesh, the error is approximately 0.003, or less than 0.5% of the total value. The adjoint solution is not computed on the final mesh because no further mesh refinement is needed to meet the error tolerance, however, it could be computed to obtain a final correction term and a conservative estimate of the remaining error.

3 Shape Trade Study This study compares the drag performance of six shroud shapes at transonic to supersonic conditions and zero angle of attack. Specifically, the baseline shape closely represents the command module flown on the Apollo missions, and consists

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of a conical shape with rounded tip. The dimensions of the baseline and alternative shapes are defined by the original Apollo command module, with total length L = 2.8788 m, measured from the tip to the cylindrical stack, and maximum radius R = 1.9558 m. The alternative shapes consist of a family of power-law shapes defined by Eq. (8) with exponents N = 0.4, 0.5, and 0.6, and two Sears–Haack shapes defined by Eq. (9) with coefficients C = 0.0 and 0.33. The Sears–Haack shapes are blended with a sphere of radius 25.4 cm to form a blunt nose that coincides with the blunt tip of the baseline conical shape, and is necessary for manufacturing feasibility and improved subsonic performance. Combined, these shapes comprise a set of realistic rocket fairing designs that are likely to exhibit improved aerodynamic performance over the baseline. Each shroud is mounted on a simplified Saturn V vertical stack and simulated at 12 points in the ascent trajectory, with Mach numbers ranging from 0.6 to 5.0. r = R (x/L) N A θ − 12 sin 2θ + C sin3 θ r =R π

(8) wher e

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Figure 4 shows dimensional pressure contours on the shroud surface and a slice through the pressure field at the Mach 2.0 trajectory point. The baseline conical shape exhibits a constant surface pressure in supersonic flight because the flow does not accelerate after the initial deflection at the tip of the shroud. The constant flow

Fig. 4 Pressure contours on a slice through the domain at Mach 2.0. (a) Baseline, (b) Sears–Haack C = 0.00, (c) Sears–Haack C = 0.33, (d) power-law N = 0.4, (e) power-law N = 0.5, (f) power-law N = 0.6, (g) legend


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velocity corresponds to a constant high pressure along the shroud. In contrast to the conical shape, the high curvature at the nose of the power-law and Sears–Haack shapes accelerates the flow and lowers pressure over most of the shroud, reducing the integrated axial pressure force. Figure 5 shows surface pressure distributions along the length of each shroud at Mach 2.0. Peak pressures exist at the leading point, and then drop as the flow approaches the base of the shroud. In contrast to the baseline, the Sears–Haack shapes have roughly linear descents from peak to minimum pressure, and the power-law shapes drop quickly then level out towards the base.

3.1 Drag Profiles Every Sears–Haack and power-law shape is superior to the conical shape at every Mach number tested in the study. The drag coefficients plotted in Fig. 6 reveal the improvements of each alternative shroud shape over the baseline. At subsonic conditions, the Sears–Haack C = 0.33 has the lowest drag coefficients, but loses this advantage in the supersonic regime. Characteristics that make a shape efficient at low speeds do not necessarily apply in supersonic flow, as shown by the trade-offs inherent in each of the shapes. There is a notable performer in the study, however. The power-law N = 0.4 shape is nearly as good as the Sears–Haack in subsonic flow and keeps this advantage in the transition through transonic and then supersonic flow. It’s maximum drag coefficient is reached at Mach 5.0, but the atmosphere has become thin at this point, and dynamic pressure has decreased dramatically.

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3.2 Drag Loss A metric known as “drag loss” is introduced to quantify shroud performance over the actual ascent trajectory defined in the Saturn V flight evaluation report (1969) [5]. Drag loss represents the total velocity decrease due to deceleration from aerodynamic drag in reference to a zero-drag body, see Martinez-Sanchez [2]. This evaluation method is advantageous because it considers vehicle weight, time, and dynamic pressure in addition to the drag coefficient. The shroud with the lowest drag loss will enable larger payloads and greater flexibility in launch and orbital trajectories. To find drag loss, the dimensional drag force on the body is converted to an acceleration by dividing by the vehicle’s instantaneous mass. This is then integrated over the flight time to obtain a velocity term. The computed drag loss is then a measure of the shroud’s overall effect on the final velocity of the rocket. For the discrete evaluation of drag loss, a trapezoidal integral approximation is used, hence values for dimensional drag D and mass m are averaged between trajectory points i at time ti and ti+1 .

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improvement over the baseline, and the power-law shapes are between 20 and 33% better. A trend in the power-law shapes indicates that the blunter (i.e. N = 0.4) shapes have better overall performance over the launch trajectory.

4 Conclusion Benefits of the adjoint-based adaptive meshing algorithm have been illustrated in flow conditions from subsonic to supersonic. The algorithm detects functional sensitivity to the mesh and generates an optimal mesh for the given geometry and flow conditions. Localized refinement improves the functional accuracy while maintaining the computational affordability of a large database of simulations. This technique was applied to a set of axis symmetric shroud shapes for the Saturn V launch vehicle, which are evaluated for their drag performance across the Saturn V launch trajectory. Then, a quantitative comparison of each shape’s drag performance was made with a benchmark quantity called drag loss. The most blunt power-law shape (N = 0.4) has the lowest drag loss, with a 33% improvement over the baseline shroud. This shape trade study has demonstrated how significant improvements can be achieved for both fluid dynamic computations and future rocket designs. Acknowledgements The authors would like to thank Drs. Mike Aftosmis, Marian Nemec, and Shishir Pandya for their support.

References 1. Aftosmis, M., Berger, M., Adomavicius, G.: A Parallel Mulitlevel Method for Adaptively Refined Cartesian Grids with Embedded Boundaries. AIAA-2000-0808 (2000) 2. Martinez-Sanchez, M.: Orbital Mechanics: Review Staging. From MIT OpenCourseWare lecture notes, 16.512 Rocket Propulsion, Lecture 32, Fall 2005 (2005)

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3. Nemec, M., Aftosmis, M.: Adjoint-Based Adaptive Mesh Refinement for Embedded Boundary Cartesian Meshes. AIAA-2007-4187 (2007) 4. Nemec, M., Aftosmis, M., Wintzer, M.: Adjoint-Based Adaptive Mesh Refinement for Complex Geometries. AIAA-2008-0725 (2008) 5. Saturn V Flight Evaluation Working Group.: Saturn V Launch Vehicle Flight Evaluation Report. AS-503 Apollo 8 mission (February 1969)

Mesh Quality Effects on the Accuracy of Euler and Navier–Stokes Solutions on Unstructured Meshes Aaron Katz and Venkateswaran Sankaran

Abstract We examine discretization error for standard node- and cell-centered schemes using the Method of Manufactured Solutions. We find that for isotropic grids, node-centered approaches produce less error than cell-centered approaches for comparable cell size. In contrast, cell-centered schemes produce less discretization error on stretched meshes. In 3D, careful treatment of non-planar faces is necessary to avoid first-order errors. We introduce a new corrected scheme which enhances the accuracy of the node-centered scheme to third-order.

1 Introduction The need to resolve high Reynolds number viscous flows around complex geometry places high demands on flow solvers. Unstructured schemes have acheived mainstream use for such problems due to their ability to automatically discretize complex domains [6]. As the complexity of CFD simulations increases, the need for rigorous verification becomes paramount. The difficulty of obtaining exact solutions for verification has led to the use of the Method of Manufactured Solutions (MMS) [8], which has acheived widespread use to evaluate order of accuracy [7, 9] and grid quality effects [5, 10]. In this work we evaluate the performance of node- and cell-centered schemes on a variety of cell-types. We confirm many previous findings as well as highlight additional findings, including a new third-order scheme and implications for 3D prismatic meshes.

2 Description of Schemes Tested Discretization errors are evaluated for 2D node- and cell-centered schemes as well as a 3D cell-centered prismatic scheme applied to a general convection-diffusion equation. The 2D node-centered scheme is derived from a Galerkin finite element A. Katz (B) US Army Aeroflightdynamics Directorate (AMRDEC), Moffett Field, CA 94035, USA e-mail: [email protected]


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method, with control volume face areas computed as the sum of the dual facets touching each edge. Upwinding is introduced via artificial diffusion using linear least squares reconstruction. The viscous discretization is also based on a Galerkin approximation, as described by Barth [1]. In this work, we present a new “corrected” scheme, which obtains formal thirdorder accuracy for inviscid terms only on arbitrary triangular meshes with the addition of a correction term at each node, C0 =

1 i

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(1)

where ∇ F is the gradient of the flux, computed to second-order accuracy with a quadratic least squares method. Niether second derivatives nor higher-order quadrature are needed as for cell-centered quadratic schemes [2]. While the corrected scheme appears to work well for isotropic grids, detailed analysis of the method for stretched grids is reserved for future work. In the cell-centered formulation, the state variables are located at the cell-centers, and the faces are the boundaries of the primary cells. Reconstruction is performed by first obtaining nodal values of the variables with a linear least squares procedure, followed by a Green – Gauss gradient integration around the perimeter of each control volume. The viscous terms are discretized with a combination of reconstructed nodal and cell-center values to obtain an estimate of the gradient at the face quadrature points. The 3D prismatic solver is based on the same approach. Since general polyhedral volumes in 3D are composed of non-planar faces, we implement a flux integration scheme in which each non-planar face is triangulated with a quadrature point placed at the center of each resulting triangular facet. We also implement a simplified flux quadrature method that uses only a single quadrature point at the center of each non-planar face.

3 Accuracy Tests on Isotropic and Stretched Grids Using MMS solutions shown in Fig. 1, we verify inviscid and viscous orders of accuracy using uniform and randomly perturbed grids of quadrilaterals, equilateral triangles, and right triangles. The cell size for a given mesh and scheme is defined as ds = (Vtotal /n do f )1/d , where Vtotal is the total volume of the domain, n do f is the number of degrees of freedom in the mesh (nodes or cells), and d is the number of spatial dimensions. Inviscid and viscous discretizations are tested in turn to isolate the resulting discretization errors. The results of the isotropic grid order of accuracy test indicate that triangles produce less discretization error than quads for a given cell size, shown in Fig. 2 for the inviscid discretization, and Table 1 for both inviscid and viscous discretizations. In the node-centered case, the median-dual approximation actually produces

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Node, RT, reg. Node(cor.), RT, reg. Node, RT, pert. Node (cor.), RT, pert. Node, ET, reg. Node (cor.), ET, reg. Node, ET, pert. Node (cor.), ET, pert. Node, Q, reg. Node, Q, pert.

1 1 2

3 1

500

1000 1500

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10–3 1 10–4

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Cell, RT, reg. Cell, RT, pert. Cell, ET, reg. Cell, ET, pert. Cell, Q, reg. Cell, Q, pert.

10–6 500

1000 1500

ds–1 (b)

Fig. 2 Error convergence of inviscid terms, regular and perturbed isotropic grids. (a) Nodecentered, (b) cell-centered

404


Table 1 Order of accuracy of inviscid and viscous terms as predicted by the isotropic scalar MMS test. Perturbed and regular grid results given Scheme

Quadrilateral pert./reg.

Equilateral triangle pert./reg.

Right triangle pert./reg.

Node-centered linear, inviscid Node-centered corrected, inviscid Cell-centered, inviscid Node-centered linear, viscous Cell-centered, viscous

1/2 – 2/2 2/2 2/2

2/3 3/3 2/2 2/2 2/2

2/3 3/3 2/2 2/2 2/2

first-order errors for quads, consistent with the findings of Diskin and Thomas [3]. For the cell-centered case, second-order accuracy is maintained for quads, but at a higher level of error. On the other hand, triangles always produce third-order errors for the corrected scheme, with second-order accuracy observed on arbitrary meshes for both the node- and cell-centered schemes. With the proper Galerkin-weighted MMS source term, the linear node-centered scheme show third-order accuracy, contrary to the findings of Diskin. The proper discretization of the MMS source term appears to be critical for correct prediction of orders greater than two [7]. Due to the third-order accuracy on regular triangles, along with the unconditional third-order accuracy of the corrected scheme, nodal schemes appear best suited for isotropic grids, as long as quads with median-dual approximations are avoided. All viscous schemes tested result in second-order accuracy for with comparable levels of error. Along with isotropic grids, the performance of the schemes on stretched meshes of various types is tested. A total of four mesh types are used, including flat and curved surface boundary layer type meshes, shown in Fig. 3, along with the corresponding manufactured solutions shown in Fig. 1c, d. Three levels of mesh stretching are tested, including wall cell aspect ratios of 102 , 104 , and 106 . As in the isotropic tests, inviscid and viscous discretizations are tested independently to isolate the effects of mesh stretching on each discretization type. The curved surface error convergence for right triangles is shown in Fig. 4. Similar trends are seen for the inviscid terms and for the flat plate case. The slopes in Fig. 4a indicate the cell-centered scheme produces less discretization

Fig. 3 Flat and curved quad and triangular stretched grids used to assess grid quality effects


405 Node, RT, A = 102 Node, RT, A = 104 Node, RT, A = 106 Cell, RT, A = 102 Cell, RT, A = 104 Cell, RT, A = 106

100

L2 error

10–1

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2 1

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1000 1500

ds–1 (a) Cell, RT, A = 106 Cell, Q, A = 106

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10–1

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1000 1500

ds–1 (b)

Fig. 4 Comparison of error convergence of viscous terms for the curved boundary layer case. (a) Node- and cell-centered rt. triangles, (b) quad and rt. triangular cell-centered

error than the node-centered scheme independent of aspect ratio. Furthermore, Fig. 4b shows that quads produce significantly less error for a given cell size than triangles for stretched meshes. Cell-centered quads of aspect ratio 106 produce less error than node-centered triangles of aspect ratio 102 . These results are in contrast to the isotropic results, which favor node-centered schemes on triangular grids.

406


4 Implications for Non-planar Faces in 3D The low errors produced by the cell-centered scheme on quads motivate the use of cell-centered prismatic meshes in 3D. Here, we present an order of accuracy study regarding the non-planar control volume faces characteristic of 3D prismatic grids. Regular and perturbed prismatic grids are studied, exhibiting planar and nonplanar faces, respectively. Single point and triangulated face quadrature methods are evaluated. With single point quadrature, the convective flux is evaluated once at the face center of each quadrilateral face. With triangulated face quadrature, each quadrilateral face is first triangulated, with a quadrature point placed at the center of each triangular facet for a total of two quadrature points per quadrilateral face. The results of the study are shown in Fig. 5 and summarized in Table 2. The single-point quadrature is unable to retain second order accuracy in general. In contrast, the triangulated face quadrature maintains second-order accuracy, even on perturbed grids with non-planar faces. This result is either largely unknown in the literature, or largely ignored, as pointed out by Delanaye and Liu [2]. However, face triangulation was also advocated in the work of Liu and Vinokur [4].

10–6 1 1

L2 error

10–7

10–8

10–9

single pt. quad., reg single pt. quad., pert. triangulated face, reg. triangulated face, pert.

1 2

10–10 ds–1

20

40

60 80

Fig. 5 Error convergence showing first-order effects of simple flux integration on non-planar faces for 3D prismatic grids

Table 2 Order of accuracy for planar and non-planar faces for cell-centered prismatic grids Quadrature scheme

Planar faces

Non-planar faces

Single point Triangulated face

2 2

1 2


407

5 Conclusions The MMS procedure is used to predict order of accuracy precisely and evaluate mesh quality effects for a variety of cell types and schemes. Nodal approaches, including a new corrected scheme, produce less error than cell approaches on isotropic grids, showing third-order accuracy on triangles in many cases. However, the median-dual approach on quads produces first-order errors. In contrast, cellcentered schemes produce less error on stretched meshes, especially for quads. The stretched mesh results encourage the use of cell-centered prismatic meshes in 3D for high Reynolds number flows. In 3D, first-order errors arising from non-planar faces are observed. Second-order accuracy is recovered by triangulation of non-planar faces. Future work will focus on the effects of curvature and high aspect ratio in 3D. Acknowledgements Development was performed at the HPC Institute for Advanced Rotorcraft Modeling and Simulation (HIARMS), supported by the DoD High Performance Computing Modernization Office (HPCMO). Material in this chapter is a product of the CREATE-AV Element of the Computational Research and Engineering for Acquisition Tools and Environments (CREATE) Program sponsored by the U.S. Department of Defense HPC Modernization Program Office.

References 1. Barth, T.J.: Numerical aspects of computing viscous high reynolds number flows on unstructured meshes. AIAA Paper 1991-0721, AIAA 29th ASM, Reno (1991) 2. Delanaye, M., Liu, Y.: Quadratic reconstruction finite volume schemes on 3d arbitrary unstructured polyhedral grids. AIAA Paper 1995–3259, AIAA 14th CFD Conference, Norfolk (1999) 3. Diskin, B., Thomas, J.: Accuracy analysis for mixed-element finite-volume discretization schemes. NIA Report 2007–08, National Institute of Aerospace (2007) 4. Liu, Y., Vinokur, M.: Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. J. Comput. Phys. 140, 122–147 (1998) 5. Luke, E., Hebert, S., Thompson, D.: Theoretical and practical evaluation of solver-specific mesh quality. AIAA paper 2008-0934, AIAA 46th ASM, Reno (2008) 6. Mavriplis, D.J.: Unstructured mesh discretizations and solvers for computational aerodynamics. AIAA paper 2007-3955, AIAA 18th CFD Conference, Miami (2007) 7. Pautz, S.: Verification of transport codes by the method of manufactured solutions: The attila experience. Tech. Rep. LA-UR-01-1487, Los Alamos (2001) 8. Roache, P.: Code verification by the method of manufactured solutions. Trans. ASME 124, 4–10 (2002) 9. Roy, C.: Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205:131–156 (2005) 10. Sun, H., Darmofal, D., Haimes, R.: On the impact of triangle shapes for boundary layer problems using high-order finite element discretization. AIAA Paper 2010-0542, AIAA 48th ASM, Orlando (2010)

Analysis of a RK/Implicit Smoother for Multigrid R.C. Swanson, E. Turkel, and S. Yaniv

Abstract The steady-state compressible Navier – Stokes equations are solved with a finite-volume, second-order accurate scheme. The equations are solved with a multigrid algorithm that uses a 3-stage Runge – Kutta scheme with an implicit preconditioner as a smoother. We analyze this smoother in which the implicit system is approximately inverted by a few symmetric Gauss – Seidel relaxation sweeps. The analysis for the linear system determines the Fourier spectrum of the multigrid smoother. Improved performance of the algorithm based on the analysis is demonstrated by computing laminar flow in a rocket motor and turbulent flow over a wing.

1 Introduction Multigrid algorithms with an explicit Runge – Kutta (RK) scheme and implicit residual smoothing (IRS) are the foundation of many existing aerodynamic prediction codes. This class of methods has recently been significantly improved by replacing the scalar form of IRS with a matrix form. The matrix form allows a large CFL number, resulting in faster propagation of information (enhancing multigrid efficiency since the Navier-Stokes equations contain a hyperbolic part) and reduced discrete stiffness. In this chapter we analyze this multigrid smoother (RK/implicit scheme), introduce improvements in efficiency, and demonstrate the robustness of the modified algorithm.

2 RKI Algorithm To discretize the governing fluid dynamic equations we apply a cell-centered, finitevolume approach. The advection terms are approximated with three point central differencing plus numerical dissipation. A matrix-valued or Roe-type dissipation is applied such that the scheme is second order in smooth regions of the flow field and R.C. Swanson (B) NASA Langley Research Center, Hampton, VA 23681, USA e- mail: [email protected]


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first order in the neighborhood of shock waves. The viscous terms are discretized with a second-order central difference approximation. See [1, 5] for details. For solving the discrete equations we use the full approximation storage (FAS) multigrid algorithm described in [5]. This algorithm uses full coarsening and either a V-type or W-type cycle. The operators for restricting residuals and flow variables to the coarser grids are determined by a conservative residual summation and volume weighting, respectively. Coarse grid corrections are transferred to finer grids by linear interpolation operators. The RK scheme of the multigrid smoother has three stages with coefficients [α1 , α2 , α3 ] = [0.15, 0.4, 1.0]. The solution vector W on the q-th stage of the RK scheme is given by W(q) = W(0) + δW(q) = W(0) − αq

t L W(q−1) = W(0) − αq t R(q−1) , (1) V

where L is the complete difference operator for the system of equations, t is the time step, V is the volume of the mesh cell being considered, and R(q−1) represents the residual function for the (q −1)-th stage. A general form for the residual function can be written as R

(k)

1 1 = L W(k) = V V

3 (k)

L W c

−

k r =0

v

γkr L W

(r )

+

k

4 γkr L W d

(r )

, (2)

r =0

v , and L d denoting convective, viscous, and dissipative operators. For with L c , L consistency γkr = 1. The coefficients γkr are the weights of the viscous and dissipative terms on each stage, and for the 3-stage scheme,

γ00 = γ 1 ;

γ10 = 1 − γ 2 , γ11 = γ 2 ;

γ20 = 1 − γ 3 , γ21 = 0, γ22 = γ 3 . (3)

When the weights [γ 1 , γ 2 , γ 3 ] are [1.0, 1.0, 1.0], this is called standard weighting. Based upon analysis and numerical testing we have determined that the modified weights [1.0, 0.5, 0.5] lead to improved robustness of the smoother. In particular, the Mach number used in calculating the preconditioner needs to be cutoff to prevent a zero Mach number. The modified weights allow a lower level for these cutoffs which leads to a faster rate of convergence. Letting Li be an implicit operator, we define the following replacement for the explicit update in Eq. (1): (q)

δW

= −αq

(q−1) t t Fn S, P L W(q−1) = −αq P V V

(4)

all faces

where P is the implicit preconditioner defined by the approximate inverse LBi−1 , Fn is the normal flux density vector at the cell face, and S is the area of the cell face. A first-order upwind approximation based on the Roe scheme is used for the convective derivatives in the implicit operator, which is defined by

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3

411

4 t (q−1) t (q) I +ε An S δW = −αq Fn S. V V all faces

(5)

all faces

The matrix An is the flux Jacobian associated with Fn at a cell face, and ε is an implicit parameter. Using one-dimensional Fourier analysis and numerical testing we have found that a good choice for ε is 0.5. An approximate inverse of the implicit operator for the linear system is obtained with Gauss-Seidel (GS) relaxation. Complete discussion of the scheme is given in [3, 6].

3 Fourier Analysis To analyze the RK/implicit scheme we consider a finite domain with periodic boundary conditions. We discretize the linearized, time-dependent Navier-Stokes equations on a Cartesian grid with m f × n f cells covering the domain. For convenience, the Navier-Stokes equations are transformed from conservative variables to primitive variables. We let U j1 , j2 denote the discrete solution vector of primitive variables that resides at the mesh point ( j1 h x , j2 h y ). Then, the preconditioned form of Eq. (1) is Fourier transformed to obtain DLCh U C(q−1) . C(q) = U C(0) − αq t P U k1 ,k2 k 1 ,k 2 k 1 ,k2 V

(6)

Lh is the linearized discrete residual operator for the primitive variables. The transformed discrete vector function is given by (q) C Uk1 ,k2 =

m f −1 n f −1 1 (q) −i(j1 θx +j2 θ y ) Uj1 ,j2 e , mf nf

(7)

j1 =0 j2 =0

where the phase angles θx and θ y , along with the corresponding wave numbers are given by k1 θx = 2π , mf

1 k2 1 mf − 1 , · · · , mf , θ y = 2π , k1 = − nf 2 2 1 1 nf − 1 , · · · , nf . k2 = − 2 2

(8)

The transformed residual operator LCh is a function of the transformed shift operators, which are defined by Cx ≡ eiθx , E

Cy ≡ eiθ y , E

−π < θx ≤ π,

−π < θ y ≤ π.

(9)

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After applying the 3-stage RK scheme with standard weighting of dissipation, we have Cn C Un+1 k1 ,k2 = Gr ki Uk1 ,k2 ,

(10)

where the amplification matrix DLCh + α 2 α 1 (P DLCh )2 − α 3 α 2 α 1 (P DLCh )3 , Gr ki = I − α 1 P I is the identity matrix and α q represents the product of the RK coefficient and the ratio of t to V . Pierce and Giles [2] introduced a matrix preconditioner determined by the diagonal elements of the residual function. This is equivalent to a block Jacobi preconditioner. In the RKI scheme we introduce off-diagonal terms. An approximate inverse of the resulting implicit system is obtained using a small number (usually two) of symmetric Gauss-Seidel sweeps. This method can be considered a block Gauss – Seidel preconditioner. In Fig. 1 we show the Fourier footprints for the RK(3,3) scheme (3 stages, 3 dissipation evaluations) with block Jacobi and block GS (i.e., RKI) preconditioners. The Fourier footprints are eigenvalue distributions corresponding to the Fourier symbol of the operator V −1 PL . For these plots the Mach number is M = 0.5, the Reynolds number is Re = ∞, and there is flow alignment (flow angle α = 0◦ ). There is a moderate mesh cell aspect ratio (A R) of five. The dashed and solid lines in the figures are the absolute stability curves with standard and modified dissipation weighting. The eigenvalue distributions are associated with all modes having a highfrequency component in at least one direction. There are four eigenvalue footprints, which correspond to the two convective (entropy and vorticity) modes and the two acoustic modes. The entropy footprint is either above or below the real axis. The two acoustic footprints are above and below the real axis, and the vorticity footprint lies between them. With block Jacobi there is poor damping of certain high-frequency modes corresponding to acoustic and convective modes. First-order upwind differencing is used for the residual function, but the character of the clustering does not change with second order. The block GS preconditioner provides good damping except for certain convective modes, and this is a consequence of the flow alignment that results in a vanishing eigenvalue. Most eigenvalues are well clustered away from the origin of the complex plane, which is necessary for effective damping. By introducing an appropriate entropy fix, all modes are effectively damped with block GS. Figure 2 displays contours of the spectral radius of the amplification matrix for RKI(3,3) when the Fourier angles −π ≤ θx , θ y ≤ π with a flow angle 45◦ . Two symmetric Gauss-Seidel (SGS) sweeps are used to approximate the inverse of the implicit system. High-frequency components lie outside the dashed line square. In this comparison of damping behavior of the RKI(3,3) scheme with standard and modified dissipation weighting, the improved damping, especially of the highest frequencies, with modified weighting is evident. The modified weighting provides the additional advantage of extending the absolute stability curve of the RKI(3,3)

Analysis of a RK/Implicit Smoother for Multigrid 3

3

wgts: [1 1 1] 2

wgts: [1 1 1] 2

wgts: [1 .5 .5]

1 Im

413

wgts: [1 .5 .5]

1 Im

0 –1

0 –1

–2

–2

–3 –10

–8

–6

–4

–2

–3 –10

0

Re

–8

–6

–4 Re

(a)

(b)

–2

0

0.

1

0.2

θy

0.3 0.4

0.2

1

4 0. 0. 6 8

25

0 0 .4 0. .6 8

1 0.

0.

8

0.35

θy

0.

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0. 0.15

0.2

2

0.

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0.

2

1

1 0

2

0.

0.15

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0.

2

0.

5

2

0.2

0.

0.2

25

0.35

5 0.2

0.

0.2

1

0.

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35

Fig. 1 Fourier footprints of RK(3,3) scheme with two preconditioners for all modes with highfrequency components (64 × 64, M = 0.5, α = 0◦ , CFL = 103 , A R = 5, Re = ∞). (a) Block Jacobi (residual: 1st order), (b) Block Gauss – Seidel (residual: 2nd order)

–1

–2

–2

15

0.

8

0

–1

0.

–3

–3 –3

–2

–1

0

θx

(a)

1

2

3

–3

–2

–1

0

θx

1

2

3

(b)

Fig. 2 Effect of dissipation weights on damping behavior of RKI(3, 3) scheme (64 × 64, M = 0.5, α = 45◦ , CFL = 103 , A R = 1, Re = ∞, 2 SGS). (a) wgts: [1, 1, 1], (b) wgts: [1, 0.5, 0.5]

scheme by a factor of two along the negative real axis in the complex plane, as seen in Fig. 1. This benefit can be quite important when more dissipation is introduced into the scheme, which can occur, for example, when there is a significant increase in the eddy viscosity of a turbulent flow. Moreover, it results in a higher level of robustness for the flow solver. Figure 3 shows the influence of two different values of Re, and representative corresponding values of A R, on the damping behavior of the RKI(3,3) scheme. We observe a change in the character of the damping curves due to the A R. There is still good smoothing of all high-frequency modes, and the smoothing factors for Re = 102 and Re = 106 are approximately 0.27 and 0.43, respectively. The good

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3

0.2

0.25

2

2

0.2

θy

.

0.25

0

1

0.4

0.9

0.15

.

θy

1

0.2

0.15

5

1 0.

0.35

0

0.25

0.95

–1

–1

0.2

–2

0.15

0.2

–2

0.25

0.2 –3

–3 –3

–2

–1

0

1

2

3

. –3

–2

–1

0

1

2

3

θx

θx

(a)

(b)

Fig. 3 Damping behavior of RKI(3,3) scheme with variation in Re and AR (64 × 64, M = 0.5, α = 45◦ , 2 SGS). (a) Re = 102 , A R = 10, C F L = 103 , (b) Re = 106 , A R = 103 , C F L = 104

smoothing factors for the various flow parameters indicate that the RKI(3,3) scheme is suitable for full-coarsening multigrid.

4 Numerical Results To demonstrate the robustness of the present scheme we show a result for an axisymmetric solid rocket motor. The solid rocket motor flowfield is characterized by a high temperature and a very slow speed (M < 0.1) inside the chamber. The flow accelerates towards the nozzle throat and a highly supersonic flow develops in the diverging portion of the nozzle. Figure 4a shows the full range of Mach contours in the chamber and nozzle. In Fig. 4b we zoom on the Mach values inside the motor where the sound speed is around 1,000 m s−1 and the Mach values are low. 0.15

0.15

0.05

0.1

Y

2.1 1.7 1.3 0.9 0.5 0.1

0.1

Y

M 0.1 0.08 0.06 0.04 0.02 0

M

0.05

0

0

–0.05

–0.05 0

0.1

x

(a)

Fig. 4 Rocket motor

0.2

0.3

0

0.2

0.1 x

(b)

0.3


415

The flow is solved using the Navier-Stokes equations for laminar steady flow. This computation was done using a three-dimensional (3-D) code with a single cell in the circumferential direction. An upwind scheme with a Van Albada limiter was used to discretize the flow equations. The RKI(3,3) scheme with three levels of multigrid was used to solve the flow equations. One boundary condition was injection of flow on the propellent surface according to Vielle’s law m˙ = ρ p a

p pref

n

where m˙ is the injected mass flux, ρ p is the propellent density, a is the burning rate, and n is the burning rate exponent. The other boundary conditions were as follows: no slip on the chamber and nozzle walls, extrapolation on the supersonic motor exit plane. In Fig. 5 we compare the convergence behavior of the multigrid scheme using RKI(3,3) with that using the original (scalar) implicit residual smoothing [5]. A well converged solution is obtained with the matrix smoother (CFL = 1000), whereas the computation with the scalar smoother (CFL = 2.5) exhibits a much slower convergence. To verify the effect of the new dissipation weights [1.0, 0.5, 0.5] in the RKI(3,3) scheme, we computed transonic flow over an ONERA M6 wing with a free-stream M∞ = 0.83 and Re = 2.1158 × 107 using a Baldwin – Lomax turbulence model. A matrix-valued dissipation was used [4]. Entropy fixes were applied to the eigenvalues of the preconditioning matrix in Eq. (5). For the entropy fix of the convective

101 100

implicit smoothing RK/implicit smoothing

log(err)

10–1 10–2 10–3 10–4 10–5 10–6 10–7 0

200

400

Fig. 5 Convergence rate for rocket motor

600 800 cycles

1000 1200 1400

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Table 1 Convergence for ONERA M6 wing for various dissipation weights and cutoff parameters Dissipation weights Cutoff parameters Residual Conv. rate 1.0, 0.5, 0.5 1.0, 0.5, 0.5 1.0, 0.5, 0.5 1.0, 0.5, 0.5 1.0, 0.5, 0.5 1.0, 1.0, 1.0 1.0, 1.0, 1.0 1.0, 1.0, 1.0 1.0, 1.0, 1.0 1.0, 1.0, 1.0

.10, .20 .10, .10 .08, .05 .08, .04 .06, .04 .10, .20 .10, .10 .08, .05 .08, .04 .06, .04

.327 × 10−4 .445 × 10−5 .472 × 10−6 .230 × 10−6 .182 × 10−6 .348 × 10−4 .500 × 10−5 .604 × 10−6 NC NC

.8798 .8619 .8425 .8364 .8344 .8937 .8631 .8447

eigenvalues we use a two-parameter function of the mesh aspect ratio. The two parameters multiply the bounds on these eigenvalues. If these parameters are chosen too large, the convergence slows down. When they are too small, the scheme can become unstable. Hence, we want these parameters to be as small as possible without destroying stability. Since this function is nonlinear, it cannot be modeled with the Fourier analysis. The two parameters are very dependent on the dissipation weighting used. In Table 1 we present the convergence after 100 multigrid V-type cycles; NC means no convergence. We see that the convergence for fixed cutoff parameters with the weights [1.0, 0.5, 0.5] is slightly better. What is more important is that the new dissipation weights make the scheme more robust, allowing lower cutoffs which result in faster convergence. We have also done similar computations for two-dimensional transonic flow around the RAE 2822 airfoil. The trends were the same, although the required cutoffs were slightly larger.

5 Concluding Remarks Fourier analysis of the RKI(3,3) scheme has been considered to study the effect of various flow parameters. Using the analysis we have compared block Jacobi and block Gauss – Seidel preconditioners. The improvement of the current RK scheme, which uses block Gauss – Seidel, has been shown. The analysis has demonstrated that this scheme has good smoothing and eigenvalue clustering properties for all high-frequency error components, making it a suitable smoother for full-coarsening multigrid. In addition, the advantages of a new weighting of dissipation, including an increased reliability of the scheme, have been discussed. The implicit preconditioner introduced in [3], investigated and improved in [6], has been implemented in several three dimensional codes using either centraldifferencing with a matrix-valued dissipation or an upwind Roe scheme for the convective terms. Previously, acceleration of the convergence to a steady state has been achieved for turbulent flows over airfoils and wings and for flows in turbomachinery. In this study we have demonstrated that this scheme can be applied to flows inside rocket motors.


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References 1. Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. AIAA Paper 81–1259 (1981) 2. Pierce, N.A., Giles, M.B.: Preconditioned multigrid methods for compressible flow calculations on stretched meshes. J. Comput. Phys. 136, 425–445 (1997) 3. Rossow, C.-C.: Efficient computation of compressible and incompressible flows. J. Comput. Phys. 220, 879–899 (2007) 4. Swanson, R.C., Turkel, E.: On central difference and upwind schemes. J. Comput. Phys. 101, 292–306 (1992) 5. Swanson, R.C., Turkel, E.: Multistage schemes with multigrid for Euler and Navier-Stokes equations. NASA TP 3631 (1997) 6. Swanson, R.C., Turkel, E., Rossow, C.-C.: Convergence acceleration of Runge-Kutta schemes for solving the Navier-Stokes equations. J. Comput. Phys. 224, 365–388 (2007)

Anisotropic Adaptive Technique for Simulations of Steady Compressible Flows on Unstructured Grids O. Feodoritova, D. Kamenetskii, S.V. Kravchenko, A. Martynov, S. Medvedev, and V. Zhukov

Abstract This chapter presents the anisotropic adaptive technique applied to solution of the 2D compressible Navier–Stokes and Euler equations. The equations are discretized by the higher order finite element scheme on unstructured triangular grids. Anisotropic grid refinement is applied for grid adaptation to provide accurate computations of multiscale solutions. A heuristic Hessian-based error indicator is used to capture directional information. For finite elements of a polynomial order k > 1 p−multigrid can serve as a linear solver, or preconditioner for GMRES. For the low order k = 1 scheme the geometrical multigrid is employed based on hierarchical structures of the adaptive grid.

1 Introduction We consider the governing equations (the Euler or Navier–Stokes equations supplemented by Spalart–Allmaras turbulence model) written in the form of conservation laws. A framework of high resolution gridding technology is presented for automatic unstructured grid generation and viscous flow adaptation. The goal is to iteratively resolve strongly anisotropic behavior of solution, i.e. grid cell aspect ratio and stretching direction must follow the numerical solution anisotropy at every point in space (including boundary and shear layers, shocks etc.). At the same time the order property for the discrete approximation must be maintained, i.e. the error of numerical solution must decrease in appropriate way on a sequence of adaptively refined grids with increasing number of grid nodes. In practice, higher order finite difference schemes are favorable due to lower computational cost. However, accuracy might be lost due to highly anisotropic grids. To overcome this difficulty, we use higher order finite element (FE) scheme [7], based on the SUPG scheme [5]. In combination with the presented solution-adaptive S. Medvedev (B) Keldysh Institute of Applied Mathematics, Moscow, Russia e-mail: [email protected]


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refinement technique capable also for the dynamic grid adaptation the stabilized FE scheme provides a powerful tool both for steady and time-dependent flow modeling. Higher order accurate schemes offer potentially large savings in computational resources but might have difficulty in computing steady flow, when convergence of the discrete nonlinear steady state equations is desired. This is especially true for rather irregular and high cell aspect ratio adaptive grids, and special care must be taken to ensure the “rock-solid” convergence of the nonlinear solver. The implicit time integration is used to march the solution to steady state. On each time step we solve systems of linear equations arising from linearization of the discrete nonlinear scheme. For finite elements of a polynomial order k > 1 p-multigrid can serve as a linear solver, or preconditioner for GMRES. For the low order k = 1 scheme the geometrical multigrid is employed based on hierarchical structures of the adaptive grid.

2 Anisotropic Adaptation Adaptive generation of computational grids is a necessary requirement for accurate modeling of complex flows. It is a challenging problem that involves substantial multidisciplinary effort including approximation theory, error estimates for Navier– Stokes equations and computational geometry. The problem becomes especially complicated when the curved boundaries describing realistic geometries are to be treated. Optimal computational grids for viscous flows feature a mixture of stretched grid cells with high aspect ratio needed to resolve thin boundary and shear layers and isotropic cells for description of essentially isotropic flow features. As follows from the approximation theory, a Hessian based metric can be employed to describe optimal grid cell size and stretching direction distribution in space. The Hessian should be estimated based on numerical solution and some regularization is needed. The regularization can be directly applied to the Hessian matrices but the differences in scales between boundary layers and far-field is so large that usual smoothing procedures would significantly distort the grid optimality. Even if the problem of isotropic grid generation had been satisfactory solved for infinite space with a given metric – and it is not the case in general 3D situation when the grid quality control is essential – the presence of curved boundaries considerably aggravates the perspectives of the optimal grid generation based on the “first principles” Hessian metric. Another approach is to combine the regularization of the Hessian metric and the grid elements in real space. The concept of macro-grid introduced in [6] is a practical implementation of such an approach at the expense of deviation from the strict grid optimality but with a possibility to directly control the grid quality and optimize boundary conforming grid. An initial grid is needed to begin the adaptation procedure. Usually a coarse isotropic grid is enough for that. The combination of stretched and isotropic grid cells in the course of the adaptive grid generation is then obtained by subsequent refinement/de-refinement operations on the macro-grid

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Fig. 1 (a) Three-element airfoil. Left: macro-grid. Right: upper – fragment of macro-grid, lower – computational adaptive grid in this fragment. (b) Slotted airfoil solution. Level lines of Mach number and a fragment of the adaptive grid are shown

element hierarchy: edges and cells (also faces in 3D case) followed by macro-cell triangulation (tessellation). The resulting computational grid is used by a solver. The edge error indicator for an edge vector e |∂u/∂e| |∂ 2 u/∂e2 |1/2 |e| + w2 E t = w1 |∇u|1−qn |H |(1−qn )/2 is based on the optimal anisotropic estimates for linear interpolation error in L p norm [1, 2]. The value of qn is determined by the value of p and the dimension of space n: qn = 1 − 1/(2 p + n). The L ∞ norm corresponds to qn = 1. Lower values of qn < 1, corresponding to weaker norms, provide a possibility to redistribute the adaptive grid from sharp solution features (boundary layers and shocks) to other flow field regions. A reasonable minimal value of qn is determined by the condition p ≥ 1. The majorant function |H | for the Hessian, entering the error indicator, can be replaced by its maximal eigenvalue. A usual choice of the scalar function to adapt is the Mach number. The main purpose of the first derivative term is a regularization. The weights in the edge error indicator are adjusted in order to assure that a prescribed fraction of edges to be refined are flagged due to the second derivative term. The goal function for the mechanics of the macro-grid adaptation is to provide the equidistribution of the error indicator Et over the edges of the computational grid with a reasonable accuracy. The macro-grid and the computational grid are demonstrated in Fig. 1a.

3 Discretization Three broad classes of difference schemes for achieving higher order accuracy were reviewed and unified under a common framework. These are finite volume, stabilized finite element and Discontinuous Galerkin (DG) methods [7]. The higher order

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schemes are exceptionally efficient for smooth inviscid and viscous flows. For flows with discontinuities in the solution or its derivatives, the accuracy of all higher order schemes is degraded (asymptotically) to an order dependent on the solution smoothness. This suggests the need for local solution-based grid adaptation (h-refinement) in the vicinity of these discontinuities. The grid refinement test called the torture test was proposed. It adds grid in the worst possible location as determined by solution error. Unbounded error growth suggests potential difficulties for incorporating a particular discretization into a practical h-p refinement strategy. By applying a limiter to the gradient, the second-order finite volume scheme successfully passes the test. Other schemes that pass the torture test with bounded solution errors include the monotone DG(0) scheme and the higher order SUPG(1-2) and DG(1-2) schemes. Robust damped Newton method augmented by a time term was used for nonlinear solution converged to double precision machine-zero. The combination of the Streamwise Upwind Petrov–Galerkin/Galerkin Least Squares methods SUPG(1)/GLS(1) with the node based DG(0) was shown to reliably work with the anisotropic refinement framework described above. In Fig. 1b the steady state solution for the slotted wing is presented.

4 Matrix Solver On each time step we solve systems of linear equations arising from linearization of the discrete nonlinear scheme. Two cases are considered: higher order accurate scheme with polynomial degree k > 1 and the lower order scheme with k = 1 that corresponds to linear finite elements. For model problems with smooth solutions a polynomial degree k corresponds to the local error O(h k+1 ) in L ∞ as well as in L 1 and L 2 norms. Polynomial degree k of the FE basis functions can be specified either for each cell (providing a possibility of local p-refinement) or globally over the whole triangulation. In the case of k > 1 the choice of the FE basis affects numerical solution of linear systems. We compared standard FE Lagrange (L) and hierarchical (H) bases. In the L-basis each local element function is a polynomial of the same degree k ≥ 1. The H-basis uses the standard linear element modes which are supplemented by hierarchical polynomials of higher degrees up to k. Each expansion coefficient of grid function is associated with a reference grid node (a vertex or an edge point); for k > 2 there are also the “interior” unknowns eliminated by the static condensation procedure (exact Schur complement). We developed an iterative multilevel algorithm [3]. It is based on approximate elimination of higher order degrees of freedom and can be considered as an approximate Schur complement approach. Multigrid interpretation describes such a technique much better, and this method is now known as p-multigrid [4]. In our algorithm the fine grid contains vertices of triangulation (c-nodes) and nodes related to unknowns on edges (e-nodes). The coarse grid contains only c-nodes. The results of numerical experiments for a set of test problems (diffusion, convection-diffusion, Euler, Navier–Stokes) show capability of the proposed variant

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♦ ♦

♦

♦ ♦ - c-points

a)

b)

c)

♦♦ ♦ ♦ ♦

- r-points ♦- e-points

d)

Fig. 2 Agglomerations based on isotropic and anisotropic macro-grids

of p-multigrid. In most considered problems p-multigrid achieves convergence rate independent of k and h in the case of hierarchical finite elements. For low order scheme with k = 1 the main difficulties, associated with the use of multigrid algorithm on unstructured grids, lie in the generation of coarser levels. Most of the known approaches are not suitable for anisotropic grids. For unstructured anisotropic grids, we propose a specially designed method to generate the set of coarse levels and corresponding multigrid operators, i.e. prolongation, restriction and coarse grid operators. This algorithm essentially uses the hierarchical macro-cell and macro-edge structures arising from the proposed adaptive process. The coarse grid is not geometrically triangular anymore – it is just a marked subset of fine grid nodes. In Fig. 2a–c some possible agglomerated grid cells (coarse nodes are shown by gray circles) arising from the hybrid triangle/quadrangle macrogrid are shown. In case of highly anisotropic grids with relatively small number of highly populated macro-cells another type of agglomeration is more appropriate (Fig. 2d). To derive the coarse grid equations we use operator-dependent interpolation, operator-based elimination of internal degrees of freedom and Galerkin coarse grid operators. The algorithm employs mainly the additive Schwartz method as a smoother based on decomposition of the computational domain into a set of subdomains. Such decomposition is usually produced in computations on multiprocessor computers. The proposed multigrid algorithm is used as a preconditioner to accelerate Krylov subspace iterations, in particular the GMRES method. For the proposed Approximate Schur Algorithm (ASA) the number of GMRES iterations is almost independent on the number of processors on middle stage of nonlinear iterations compared to the growth for the baseline Alternating Schwarz Method (ASM).

5 Conclusions The right ingredients for working adaptive Navier–Stokes solver are: anisotropic adaptive grids, stabilized FE schemes ensuring convergence on irregular grids, and advanced parallel matrix solvers. An optimal combination of the heuristic Hessian based error indicator, as providing most reliable directional information for anisotropic h-refinement, with

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theoretically backed error estimates (e.g. based on adjoint solution for functional outputs adaptation) has to be found. Concerning the higher order schemes the following considerations should be taken into account. For nonlinear governing equations overshoots in the solution are unacceptable in practice, as they may adversely impact solver convergence or even preclude the existence of a discrete solution on a given grid. Therefore, higher order schemes that allow nonphysical solution overshoots and oscillations must still be augmented with limiters before they can be employed routinely for computing steady-state Navier–Stokes solutions. As for the h-p refinement, wherein higher order schemes are combined with h-refinement, the difficulty with this approach lies in estimating the local smoothness of numerical solution, in particular, for a hyperbolic problem, where local criteria are not always sufficient. The coarse grid operators based on the hierarchical grid topology seems to be very useful for speed-up of convergence in massively parallel computations. Real gains of the adaptive solvers are expected in real life 3D applications with realistic geometries. There is still a way to go in this direction but the success of the 2D version is rather encouraging. Taking into account realistic 3D geometries and generation of adaptive anisotropic surface grids are the next big challenges for adaptive grid mechanics in 3D.

References 1. Alauzet, F., Loseille, A., Dervieux, A., Frey, P.: Multi-dimensional continuous metric for mesh adaptation. In: Pébay, P.P. (ed.) Proceedings of the 15th International Meshing Roundtable, Birmingham, Alabama, USA, 17–20 September 2006, pp. 191–214. Springer, Berlin, Heidelberg, New York (2006) ISBN 978-3-540-34957-0 2. Chen, L., Sun, P., Xu, J.: Optimal anisotropic meshes for minimizing interpolation errors in Lp-norm. Math. Comput. 76, 179–204 (2007) 3. Feodoritova, O., Young, D., Zhukov, V.: Iterative algorithms for higher order finite element schemes. Russ. J. Math. Modelling 16(7), 117–128 (2004). [In Russian] 4. Helenbrook, B., Mavriplis, D., Atkins, H.: Analysis of p-multigrid for continuous and discontinuous finite element discretizations. AIAA Paper 2003–3989 (2003) 5. Hughes, T.: Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations. Int. J. Numer. Meth. Fluids 7, 1261–1275 (1987) 6. Martynov, A., Medvedev, S.: A robust method of anisotropic grid generation. Grid Generation: Theory and Applications, pp. 266–275. Computing Centre RAS, Moscow (2002) 7. Venkatakrishnan, V., Almaras. S., Kamenetskii, D., Johnson, F.: Higher order schemes for the compressible Navier-Stokes equations. Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, Orlando, FL, 23–26 June 2003

Part XVI

Aeroacoustics

Euler – Navier–Stokes Coupling for Aeroacoustics Problems Michel Borrel, Laurence Halpern, and Juliette Ryan

Abstract In this talk is presented a new method to couple Euler and Navier–Stokes solvers for aeroacoustics applications based on a space-time domain decomposition technique. First results are shown for 2D laminar configurations such as the lowReynolds subsonic flow around a cylinder.

1 Introduction In the last few years numerical aeroacoustics have undergone a very rapid evolution due to the emergence of direct simulations of sound emitted by flows, and in particular turbulent flows. However, these computations raise the question of the coupling between LES and acoustics or in other words between the Navier–Stokes and the Euler computations, especially if we want to take into account the multiscale aspect, both in space and time, of the problem. What we propose here is a new method of coupling between aerodynamics and acoustics. It is based on earlier work on Schwarz waveform relaxation methods, developed for time-dependent advection – diffusion – reaction. These methods are domain decomposition algorithms. They consist in solving the equations alternatively in each subdomain, and transmitting the necessary information through differential space-time transmission conditions, see for example [2]. They allow for different discretization in different subdomains, see [1], even in a nonconformal manner [7]. They can potentially extend to coupling different models in different zones, see [5], and are therefore well-adapted to our purpose. A first step, presented in this chapter, is to test this method, for reasons of simplicity, with a Discontinuous Galerkin scheme, for (a) the CFD where the Navier– Stokes are solved and (b) the CAA where Euler, perturbed Euler or linearized Euler

M. Borrel (B) ONERA, FR-92322, Chatillon Cedex, France e-mail: [email protected]


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equations, according to test cases are resolved. Borrel et al. [3] have formulated a new Discontinuous Galerkin scheme (EDG) for the viscous term that easily applies to either structured or non structured discretizations. Special attention must be paid to the multi-scale aspect requiring highly non conforming space-time discretization for which the discontinuous Galerkin approach is particularly well adapted. The present method will be evaluated on a 2D laminar configuration: the lowReynolds subsonic flow around a cylinder. In Sect. 2 we recall the Discontinuous Galerkin algorithm we use. In Sect. 3 we present the Schwarz waveform relaxation algorithm followed by a description in Sect. 4 of the numerical coupling algorithm. Then in Sect. 5 we present two sets of experiments: we first test the implementation of Discontinuous Galerkin into the Schwarz waveform relaxation with Robin transmission condition on the heat equation. Then we present the coupling of Navier–Stokes with Euler via the classical Schwarz waveform relaxation (i.e. exchange of Dirichlet data on the interfaces).

2 Discontinuous Galerkin Applied to the Navier–Stokes Equations DG formulation: The 2D time-dependent dimensionless Navier–Stokes equations ∂t W + ∇ · F(W ) − ∇ · D(W , ∇W ) = 0

(1)

− → where W = (ρ, ρ U , ρE) is the conservation variable vector with classical notations, F and D are the convective and diffusive flux vectors. These are solved in a domain Ω discretized by either a Cartesian or an unstructured triangular grid E Th = Ωi and the associated function space Vh , Vh = {φ ∈ L 2 (Ω) | φ/Ωi ∈ Pk }

(2)

where Pk is the space of polynomials of degree k. The DG formulation based on a weak formulation after a first integration by parts is of the form: find W h in (Vh )4 such that for all Ωi in Th , ∀φ ∈ Vh ,

5 Ωi

∂t Wh φ d x =

5

Γi (Fh

− Dh ) φ dγ −

5

Ωi (Fh

− Dh )∇φ d x. (3)

Here, the numerical fluxes Fh ,Dh and Wh are approximations of F, D and W . The inviscid fluxes F are classically determined using the HLLC ([13]) or LLF techniques; the viscous flux is computed with the EDG [3] method. Time is discretized with Shu–Osher’s [12] explicit TVD time stepping RK3.

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Note on EDG The simple idea of the EDG method is to regularize locally the discontinuous solution Wh over each edge using a reconstruction in a rectangular element R E overlapping this edge (see Fig. 1). Reconstruction is done through an L 2 projection on a DG basis in E of the same order k as the DG basis defined in the elements. For k = 2, the order of the method varies from 3.5 on a regular mesh to 2.1 on quite heterogeneous meshes (see [4]).

Fig. 1 Definition of the elastoplast element

This idea of reconstruction in a rectangular element will be used for the Euler (structured) – Navier–Stokes (unstructured) coupling.

3 Schwarz Waveform Relaxation Methods These methods are based on Schwarz domain decomposition algorithms, invented by H.A. Schwarz in 1870 [11]. In order to solve a Laplace equation in the domain Ω, it is split into two subdomains with overlap Ω1 and Ω2 , in which the equation is solved alternatively. The exchange of informations is made on the boundaries by exchange of Dirichlet values. This algorithm has been extended by P.L. Lions to nonoverlapping subdomains by using different transmission conditions, such as Robin conditions [9]. For an extension to evolution problem, we couple it to a waveform relaxation algorithm, which is an extension both of the Picard’s “approximations successives” and relaxation methods for algebraic systems, due to Lelarasmee [8]. Versions with Dirichlet or optimized Robin transmission conditions have been designed in [2]. Consider for instance the unsteady heat equation with prescribed Dirichlet boundary conditions and initial data: ⎧ ⎪ ⎨ ∂t u − ν u = f in Ω × [0, T ] u(x, 0) = u 0 (x) in Ω ⎪ ⎩ u = g on ∂Ω

(4)

where u is the temperature, ν is the constant diffusive coefficient and represents the Laplace operator. A parallel version of the domain Schwarz waveform relaxation algorithm can be written for Ω = Ω1 ∪ Ω2 as

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⎧ ∂t u k − νu k1 = f in Ω1 × (0, T ), ⎪ ⎪ ⎨ k 1 u 1 (·, 0) = u 0 in Ω1 , k = u k−1 ⎪ on Γ1 × (0, T ), u ⎪ 1 2 ⎩ u k1 = g on (∂Ω1 − Γ1 ) × (0, T ),

t y

Ω1 Γ2

Γ1

Ω2 x

⎧ k k ⎪ ⎪ ∂tku 2 − νu 2 = f in Ω2 × (0, T ), ⎨ u 2 (·, 0) = u 0 , in Ω2 , k−1 k ⎪ = u on Γ2 × (0, T ), u ⎪ 2 1 ⎩ on (∂Ω2 − Γ2 ) × (0, T ). u k2 = g

In the Robin case, define n i the unit normal exterior to Ωi . The transmission conditions become (ν∂n i + p)u ik+1 = (ν∂ni + p)u kj for (i, j) = (1, 2) or (2, 1). The parameter p is determined asymptotically as a function of the physical parameters, the size of the space-time domains, and the mesh parameters (see [2]).

4 Numerical Coupling Coupling technique (see Fig. 2): The unstructured solution in Ω1 is reconstructed on a rectangle Ω S adjacent to the common edge with a rectangular element Ω2 in the same way as the EDG technique. All the reconstructed DG coefficients are sent to the other domain so that either a Dirichlet or a Robin transmission condition can be applied. On the structured side (Ω2 ), all coefficients are sent directly without reconstruction. Domains proceed in time independently, using at first predefined interface values. At the end of the time window, domains exchange their newly computed boundary cells values for all time steps (including sub times for the Runge Kutta scheme) and a new time march is carried out with updated interface values. This iterative procedure is repeated till solution ceases to vary. This method allows for different time steps and different space interface discretization as received values from other domains can be interpolated and projected on the local time-space grid.

5 Numerical Results All computations are DG-P2 and no limiters were used. For both Cartesian and unstructured computations, we used the same functional space Vh . All triangular grids have been obtained with the freeware mesh generator Gmsh [6], Unsteady heat equation: The first test case concerns the scalar heat equation (4) and is chosen to assess the performance in terms of precision and stability of the present coupling procedure of unstructured triangular and Cartesian grids in 2D.


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Fig. 2 Definition of the coupling

Robin type transmission conditions are imposed at each time step as presented in [4]. The unsteady Navier–Stokes solver is downgraded to solve an unsteady 2D heat equation. The steady numerical solution is compared with the analytic solution for a pure heat diffusion problem on the square unit [0, 1] × [0, 1]. The exact steady solution chosen is the same as in [10] defining the Dirichlet boundary conditions

1 sinh(π x)sin(π y) + sinh(π y)sin(π x) . The initial field T exact (x, y) = sinhπ is set to zero T 0 = 0 and the numerical error between exact and approximated solutions at convergence is measured in the L ∞ norm which is probably the worst norm for DG but gives a good evaluation of diffusion for wave phenomena. The domain is split into 2 subdomains (one unstructured, the other structured), with an overlap of size h as in Fig. 2, with h the length of a boundary edge. 10−2

log(Err)

10−4

Three computations were made for values of h=1/10, 1/20, 1/40. Time windows of 5 time steps have been used with 3 Runge Kutta sub time steps. Convergence of the Schwarz (Robin) scheme is obtained in 2 to 4 iterations. On Fig. 3, convergence slope (“Robin”) is shown together with those of an ideal 3rd order scheme (“order 3”) and a 4th order one (“order 4”).

order 3 order 4 Robin

10−6

10−8

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Fig. 3 Coupling accuracy

10−1

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The TVD RK time scheme is not adapted to the computation of a steady problem, so the steady solution is obtained with about 10,000 time steps or 2,000 time windows. An average order of 3.3 is obtained showing that the coupling procedure preserves the order of the original DG schemes. Low-Reynolds number flow around a cylinder: We have chosen this basic unsteady vortical flow as a first coupling illustration: reducing the Navier–Stokes computation to a small computational domain containing the main noise sources and coupling it with an Euler computation to take into account sound propagation. This test case studies the Von Karman vortex shedding in the wake behind a cylinder in 2D. Within this configuration, supersonic pockets will grow alternatively up and down, which result in generating vortices. To simplify, only two domains are considered with, for the moment, no different grid size at the interface, but of course, our iterative method will be completely justified when the space-time multi-scale aspect will be taken into account. The computation has been run with three Schwarz sub-iterations, which allow us to converge to sixth order for each time window problem. First results on the mesh (Fig. 4a – 5,000 triangles, 2,400 rectangles) shown in Fig. 4b–d present the computed vortex shedding seen through entropy isovalues. The interface introduces no obvious spurious perturbations, which is very promising for further computations using finer grids overlapping the far-field domain.

(a) Mesh

(b) Entropy

(c) Entropy

(d) Entropy

Fig. 4 Vortex shedding behind a cylinder: mesh and entropy time evolution

References 1. D’Anfray, P., Halpern, L., Ryan, J.: New trends in coupled simulations featuring domain decomposition and metacomputing. M2AN 36(5), 953–970 (2002) 2. Bennequin, D., Gander, M.J., Halpern, L.: A homographic best approximation problem with application to optimized Schwarz waveform relaxation. Math. Comp. 78, 185–223 (2009)


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3. Borrel, M., Ryan, J.: Spectral and high order methods for partial differential equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 373–381. Springer Verlag (2010) 4. Borrel, M., Ryan, J.: The Elastoplast Discontinuous Galerkin (Edg) method for the Navier– Stokes equations. J. Comp. Phys. (2010, Submitted) 5. Gander, M.J., Halpern, L., Japhet, C., Martin, V: Viscous problems with inviscid approximations in subregions: A new approach based on operator factorization. ESAIM Proc. 27, 272–288 (2009) 6. Geuzaine, C., Remacle, J.F.: Gmsh: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Num. Meth. Eng. 79(11), 1309–1331 (2009) 7. Halpern, L., Japhet, C., Szeftel, J.: Discontinuous Galerkin and nonconforming in time optimized Schwarz waveform relaxation. Proceedings of the 18th International Conference on Domain Decomposition Methods. http://numerik.mi.fu-berlin.de/DDM/DD18/~(2009) 8. Lelarasmee, E., Sangiovanni-Vincentelli, A.L., Ruehli, A.E.: The waveform relaxation method for time-domain analysis of large scale integrated circuits. IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems, vol. CAD-1, pp. 131–145 (1982) 9. Lions, P.-L.. On the Schwarz alternating method. I. In: Roland Glowinski, Gene H. Golub, Gérard A. Meurant, Jacques Périaux (eds.) First International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, USA, pp. 1–42 (1988) 10. Puigt, G., Auffray, V., Mueller, J.D.: Discretisation of diffusive fluxes on hybrid grid. J. Comput. Phys. 229, 1425–1447 (2010) 11. Schwarz, H.A.: Über einen Grenzübergang durch alternierendes Verfahren. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 15, 272–286 (1870) 12. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988) 13. Toro, E. F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)

Computational Aeroacoustics on a Small Flute Using a Direct Simulation Yasuo Obikane

Abstract This research pursues a numerical simulation of sounds generated by musical instruments. We selected a flute blown from the side of the cylinder with the preliminary tests of a recorder, a kind of flute that is blown from the front of the cylinder. Both two and three-dimensional aeroacoustic simulations were direct in the near fields by using the compressible Navier–Stokes equations, with the exception of the boundary conditions. We aimed to see the flow details where the theoretical approach was omitted [1]. In the last work presented in ICCFD5 [2], we computed a two-dimensional recorder, but were unable to confirm the theoretical results, since the direction the recorder’s blow was orthogonal to the direction of the real flute. In the present work, (1) We used the same flute shape that was used in the theoretical analysis [1]. (2) In 2007, we utilized the weighted directional scheme, which was calibrated in the high Reynolds number. However, in the present case the flow speed was low, so we dropped the weighted scheme in the inner domain. Then, we used the non-weighted scheme in the inner domain, and a new isotropic weighted scheme at the outer boundaries. (3) In addition, we made a theoretical interpretation of the T. Poinsot and S.K. Lele’s outer boundary condition [PL condition] [3], and proposed a new form for damping terms in their equation with a turbulence modeling equation. As the preliminary computational results in the two-dimensional recorder indicate, we obtained a clear wave pattern in a larger domain than that obtained in [2]. As the flute’s results demonstrate, the streamlines showed vertical oscillations near the mouth, i.e. the result implies that a vertical velocity oscillation exists. This is consistent with the main assumption of the flute theory by M.S. Howe. For three-dimensional flute computations, we confirmed the aeroacoustics coupling frequency with a guided jet’s frequency and the basic frequency of the cylinder. The

Y. Obikane (B) Institute of Computational Fluid Dynamics, Meguroku-ku, Tokyo 152-0011, Japan; Department of Mechanical Engineering, Sophia University, Tokyo 102-8554, Japan e-mail: [email protected]; [email protected]


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realizations of the main geometrical feature were successful, since small, spherical sound patterns were clearly observed.

1 Introduction It is widely known that a wind brow makes sounds, but few understand how to create mild sounds with the instruments aeroacoustically. Aeroacoustical studies were conducted for several decades, mainly for high speed flows, but few have dealt with the low speed that is known as a resonance oscillation by the global instability and local instability of vorticities. In the present analysis, we take steps toward the realization of the three-dimensional sound of a small size flute at a low speed. In addition, we interpret the role of eddies, which control the mode of sound and the instability to understand the detail mechanism.

2 Scheme for the Low Speed There is no finite difference scheme that can cover every speed of flows. The scheme in the 2007 project used for the prediction of a recorder was weighted to adjust for the inertial subrange’s cutting frequency. The main flow direction took the derivatives as 2/3, and the derivatives taken along the oblique direction were weighted as 1/3. The new representation for the present analysis is as follows: if we treat the wave propagating at outer boundary points, the waves will propagate spherically, and all directional derivatives should be equally weighted. Thus, the representation becomes as follows: u

∂ φ = c1L x(u, φ) + c2Lζ (u, φ) + c3Lη(u, φ) ∂x

(1)

where Lη(u, φ) is defined as Lη(u, φ) = u(−φ(i + 2, j − 2) + 8(φ(i + 1, j − 1) − φ(i − 1, j + 1)) +φ(i − 2, j)) + 3abs(u)(φ(i − 2, j + 2) − 4φ(i − 1, j + 1) +6φ(i, j) − 4φ(i + 1, j − 1) + φ(i + 2, j − 2))/(12h)

(2)

where the coordinates ζ and η are sketched in Fig. 1. The test computation wave patterns in 20 × 20 meshes and 50 × 50 meshes are shown in Fig. 2. The modified isotropic scheme proved to be superior to the inertial calibrated scheme in the pattern in Fig. 3.

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Fig. 3 Computational results (rectangular mark: new scheme)

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3 An Interpretation of the Poinsot and Lele’s Condition (PL Condition) The Poinsot and Lele’s damping condition can be modified to satisfy the dynamic balance, as in Fig. 4. Since the force acting on any closed surface is given by the surface integration with the momentum equation, we have a drag force if the integral is taken on the body where the velocity becomes zero. , ∂ (3) (− p + τ i, j + Πi, j )dn Drag(t) = ∂n If we choose the surface at infinity, the meaning of the integral will be the same. If we treat the turbulent flows, the turbulence modeling equation must be adjusted to satisfy to balance of the dynamics (i.e. the Reynolds stress equation must satisfy the same equilibrium condition at a far field). Thus, the momentum equation can be evaluated as m i m j /ρ ∂(mˆ i mˆ j /ρ) ∂ mˆ i =− + ∂t ∂x j ∂x j

(4)

The symmetry form of the model is written as ∂m p m i " μ < k > u i Ψ 1 p ( f ar ) + λ < k > u i Ψ 2 p ( f ar ) ∂t +μ < k > u p Ψ 1i ( f ar ) + λ < k > u p Ψ 2i ( f ar ) By taking the integration over the outer boundaries, we have , 1 < k > (u j (μα1 + λα2) + μα3 + λα3)dσ ΔMs(t) = − Uin f

Drag

Fig. 4 Acting and reacting force diagram

Wake (momentum deficit)

(5)

(6)


439

where alphas are some invariant functions of the isotropy tensor of m p m i . Since the amplitude L1 of the wave given by the Poinsot and Lele’ s no reflective condition is given as L1 = σ ( p − pin f )c/ X L ,

(7)

and if the accuracy is sufficiently high, both the drag force and the momentum deficit at the outer boundary are balanced. The sum of extremely low speed momentum at the outer boundary must yield the equal force. If the Reynolds stress equation is adjusted at the outer boundary, the PL condition should be identical in form to the turbulence model equation in the asymptotic form. Thus, the sigma in the outer condition of the PL condition is proportional to the wave number that is the form of the modeling equation.

4 Generation of Musical Sound 4.1 The Boundary Conditions The blowing velocity at the outlet of the guide to the opening of the recorder was 5–10 m s−1 . The boundary conditions were as follows: The velocity on the body was zero, and the pressure and temperature had symmetrical conditions on the body. The outlet condition was defined as the Poinsot and Lele’s condition (i.e. the damping term was adjusted for the reference pressure). If we set the no-reflective condition, the acoustic energy produced by the jet was not accumulated, and was released at the outlet.

4.2 Two-Dimensional Cases For the recorder’s preliminary computation, we took 2000 × 1000 meshes, which had sufficient points to produce the spherical waves shown in Fig. 5. They also depicted the waves more clearly than that of the 2007s result illustrated in Fig. 6. In the 2007 results for the two-dimensional recorder, the amplitude of the vortices center was 1 mm in a small domain, as indicated in Fig. 4. In the present case, two

Fig. 5 Present fine mesh case

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Grid 2000x1000

Fig. 6 Result in 2007 recorder coarse grid : 92 × 54

time:1 100

(488, 428)

27500

Fig. 7 Two dimensional flute

vorticities were also observed for the larger domain. However, the resolution became poor, so we were unable to show the 1mm vertical oscillation visually. For the twodimensional flute, we observed that vertical oscillations initiated at an extremely early stage of the computation, as shown in Fig. 7. Thus, we can infer that the flute can more easily produce a sound than the recorder.

4.3 Three-Dimensional Cases For the three-dimensional preliminary computation results of the recorder, the volume of the air in the interface chamber was so small that the sound did not project clearly. However, for the three-dimensional flute, the interface volume of the air chamber was equal to the volume of the cylinder, which had a mass sufficiently large enough to produce a stable vortex rotation. The pressure pattern is very clear


441

Grid 100 × 100 × 120

time:37.13800

step : 299500

Fig. 8 Three dim flute (pressure pattern)

in Fig. 8. Because of the three-dimensionality, we can observe the spherical pattern even though the computational domain was small.

5 Conclusion (1) The two-dimensional recorder was actually a projection model of a threedimensional flute. Thus, the sound eventually projected. (2) For the threedimensional recorder, the interface volume was so small that it was unable to produce a strong resonance among the components (a guided jet, a pipe body, and an opening). (3) The small, dimensional flute may be easier to be played as the result of this computation. Thus, if the shape of the recorder is modified to create an increased interface volume, it will generate a sound more easily.

References 1. Howe, M.S.: Contributions to the theory of aerodynamics sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71(part 4):625–673 (1975) 2. Obikane, Y., Kuwahara, K.: Direct simulation for acoustic near fields using the compressible Navier-Stokes equation. ICCDF5 pp. 85–91 (2008) 3. Poinsot, T., Lele, S.K.: Boundary conditions for direct simulation of compressible viscous flows. J. Comput. Phys. 101, 104–129 (1992)

An Iterative Procedure For the Computation of Acoustic Fields Given by Retarded-Potential Integrals Florent Margnat

Abstract An optimized procedure for the computation of acoustic fields given by retarded-time intgrals is provided. It is written in the time-domain and for fixed sources. It is devoted to applications in which there is a large amount of source data. Thus, as many observer points are required to build the acoustic image, the resulting number of source-observer pair may cause an issue in accessing to the source-observer distance, if this quantity can not be stored in a global variable. The algorithm is validated through comparisons with reference data in the case of a simple harmonic source and in the case of the aerodynamic noise generated by the cylinder flow. The implementation of the convected Green function is also presented.

1 Introduction Flow-generated acoustic fields are often predicted by computing retarded-potential integrals which appear in the formalism of aeroacoustic analogies or wave extrapolation methods. Such procedures return the acoustic emission of unsteady flows by two steps: firstly, the flow is simulated, giving access to source quantities; secondly, those quantities are propagated until observer/listener locations. In the present contribution, the numerical implementation of a retarded-potential integral is adressed in the time-domain. An optimised method is provided for the computation of the acoustic quantity on an observer grid in order to build an acoustic picture – field. It is very useful when large size source data are considered, thus involving a large amount of source-observer distances which cannot be stored in a global variable.

F. Margnat (B) Laboratoire SINUMEF, Arts & Métiers ParisTech, 151 boulevard de l’Hôpital, 75013 Paris, France e-mail: [email protected]


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2 Principle of the Procedure 2.1 Problem Formulation A general form of aeroacoustic integrals can be written, considering a source quantity S to be integrated over a source domain D in order to compute the acoustic quantity, pressure, at an observer position x and time ta , as: pa (x, ta ) =

1 4π

D

|x − y| dy S y, ta − c0 |x − y|

(1)

where y denotes the position over the source domain and c0 is the ambient sound speed. Such expression is usually called a retarded potential integral, since, unless the source is compact, the propagation distance deviation between two source points implies their respective contributions must be collected at different emission times in order to reach the observer point at the same time. The following discrete expression is considered: N ys

ΔVy j |xi − yj | l S yj , tal − 4π pa xi , ta = c0 |xi − yj |

(2)

j

where ΔVy j is the elementary volume attached to the jth source element located at yj , and N ys is the number of source elements. If it is assumed that N ts datafiles of the source term S can be computed and sampled at Δts , one have to interpolate the |x −y | source quantity at the time ta − ic0 j in order to compute the acoustic quantity on given x and ta grids. One key point is how to organise the loops on the source elements, source fields, and in the present case, observer points. The advanced time principle ([1–4]) consists in fixing the source time first, and then radiating the contribution at the observer points at a reception time which is determined by the source-observer distance. The present algorithm is dedicated to configurations for which the source-observer distances cannot be stored in a local variable. This happens when a 2D acoustic field is computed (not only a signal at a couple of observer points) using 3D or large 2D source grids (e.g. volumic source distributions).

2.2 An Optimized Algorithm Let l be fixed such as tal = lΔts , thus defining the reception time when the acoustic picture is to be computed. Following the advanced time principle, let a source time interval be fixed, which is bounded by kΔts and (k + 1)Δts , where the source quantity is available. Consequently, the [source-observer] pairs involved at this step of the accumulation process are such that:

An Iterative Procedure For the Computation of Acoustic Fields

(l − (k + 1))c0 Δts < ri j ≤ (l − k)c0 Δts

445

(3)

where ri j = |xi − yj |. If the source location is fixed as well, (3) defines two spheres centred on it of radii r1 = (l − (k + 1))c0 Δts and r2 = (l − k)c0 Δts . Thus, in two the observer plane, defined by y3 = 0, observer points verifying (3) are within

circles centred on (x1 = y1 , x 2 = y2 ) of radii r1, = r12 − y32 and r2, = r22 − y32 These geometric relations are sketched in Fig. 1-left. If the grid in the spanwise direction is basically an extrusion of the grid in the (y1 , y2 ) plane, the procedure is the following:

1. fix (loop on) the source field(s) 2. fix the reception time(s); the propagation distances r1 and r2 are computed. 3. fix (loop on) the source location(s) in the spanwise direction; the radii r1, and r2, in the observer plane are computed. 4. fix (loop on) the source coordinates (y1 , y2 ), search the observer points located within the two circles, and add the interpolated source contribution to the acoustic pressure at them. The search procedure at the fourth step is performed as follows: splitting the ring in 4 arcs, and for each grid step in the x1 direction, finding the grid steps in the x2 direction satisfying (3). As sketched in Fig. 1, for a given step i 1 such as x1 = be introduced, bounded by (i 1 −1)Δx1 ,a loop on the second observer coordinate can ⎡ ⎡ ⎤ ⎤ , 2 2 (r1 ) + (x1 − y1 ) (r2, )2 + (x 1 − y1 )2 ⎦ + 2 and i (2) = int ⎣ ⎦+1 i 2(1) = int ⎣ 2 Δx2 Δx2

Fig. 1 Left: sketch of the configuration for the implementation in the spanwise direction. M is the source point, located in (y1 , y2 , y3 ); (x1 , x2 ) are the coordinates in the observer plane; r1 and r2 are the propagation distances defined by (3); r1, and r2, are the corresponding radii of the circles obtained where the spheres centred on the source point cut the observer plane. Right: Principle of the extraction of the observer points located on the ring. Δx 1 and Δx2 are the grid steps in the observer plane

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3 Test-Case: Monopole Source The implementation of the procedure is validated through the case of a harmonic source located at the origin in a two-dimensional configuration. In the time domain, the general solution of the 2D inhomogeneous wave equation is: 1 p(x1 , x2 , t) = − 2π

+∞ +∞ −∞

⎛ ⎝

−∞

+∞ −∞

⎞ H (t − τ − r/c0 ) S (y1 , y2 , τ ) dτ ⎠ dy1 dy2 (t − τ )2 − r 2 /c02 (4)

6 where H is the Heavyside function and r = |x − y| = (x1 − y1 )2 + (x2 − y2 )2 . It is relatively straightforward to convert the time integral into an integration over a third spatial direction, noted y3 . One obtains:

⎛ 1 p(x 1 , x2 , 0, t) = 2π

⎜ ⎜ ⎜ ⎝

+∞ +∞ ⎜ 0

−∞

−∞

S y1 , y2 , 0, t −

r 2 +y32 c0

r 2 + y32

−∞

⎞ ⎟ ⎟ dy3 ⎟ ⎟ dy1 dy2 ⎠ (5)

0.15

0.15

0.1

0.1

0.05

0.05 p / (Adσ)

p / (Adσ)

This formulation shows how the procedure developed here can be validated or applied in a 2D configuration, provided that a 3D Green function is combined with a space integration over the spanwise direction and a replication of the source data known in the (y1 , y2 ) plane. The acoustic pressure computed with the algorithm presented above is compared to the reference solution obtained by computing directly (5). This allows to validate the implementation of the various loops, of the interpolation and of the integration operation. As plotted in Fig. 2-left an excellent agreement is found. 30 points by period were used to discretize the source signal, and the interpolation is linear.

0

0

−0.05

−0.05

−0.1

−0.1

0

1

2

3 x1/λ

4

5

−4

−3

−2

−1

0 x1/λ

1

2

3

4

Fig. 2 Acoustic pressure field for a monopole source in 2D. Solid line: reference solution; Symbols: algorithm solution; dashed line: r −0.5 laws. Left: quiescent medium Green’s function; Right: convected case

An Iterative Procedure For the Computation of Acoustic Fields

447

For the propagation in a uniformly moving flow at the subsonic Mach number M0 in the direction y1 , the 3D integral solution for the acoustic pressure is: pa (x, ta ) =

1 4π

D

rβ − M0 (x1 − y1 ) dy S y, ta − c0 β 2 rβ

(6)

β 2 = 1 − M02 , rβ = (x1 − y1 )2 + β 2 (x2 − y2 )2 + (x3 − y3 )2 rβ − M0 (x1 − y1 ) and τ ∗ = c0 β 2 According to the present methodology, one have to fix the reception time t and a pair of emission times τ and τ + dτ . Then one have to find the source-observer pairs for which the propagation time τ ∗ satisfies: where

(l − (k + 1))c0 Δts
0.5. If we mix cell C with a target cell Ct , so that the final valuew of the two cells is equal, α Ct wCt − wC and MCt C = we have to exchange the quantities MC Ct = αC +α Ct αC αC +αCt wC − wCt , and we easily check that wC + MC Ct = wCt + MCt C . In the 2D case, we have to make a choice for the target cell Ct . We fix Ct to be the fully-fluid cell (αCt = 0) nearest to cell C , such that the path between the two cells does not cross a solid boundary. A recursive subroutine finds such a target cell after few iterations. Let us note that the mixing procedure is entirely conservative, and ensures that the significant volume for a cell is consistent with the usual CFL condition based on standard cell size.

3 Theoretical Results The following results were theoretically proven: • Mass, momentum and energy conservation: When there is no inflow into or outflow from the domain, conservation of fluid mass is ensured. For periodic boundary conditions, the momentum is exactly balanced between fluid and solid during each time-step. For periodic or reflecting boundary conditions, the energy received by the fluid from the solid is exactly balanced by the work of fluid pressure forces on the solid during each time-step.

A Conservative Coupling Method

477

• Frame indifference: Let an arbitrary shaped rigid body moving at constant velocity and no rotation, be immersed in a uniform fluid flowing at the same velocity. The uniform motion of both the solid and the fluid is preserved by the coupling algorithm. • Free slip along a straight boundary: A uniform flow parallel to a rigid semi-infinite half-plane is preserved by the coupling algorithm. The last result shows that no numerical boundary layer or artificial boundary roughness appear at the solid boundary, even when it is not aligned with the Cartesian mesh.

4 Numerical Results Due to space limitations, we only present here a moving boundary benchmark that was first proposed in [7] and also treated in [8]. A rigid cylinder of density 7.6 kg m−3 , initially resting on the lower wall of a two-dimensional channel filled with air at standard conditions, is driven and lifted upwards by a Mach 3 shock wave. The results obtained on a 1,600 × 320 grid are shown in Fig. 1. We observe good agreement with the results shown in [2, 8]. Small differences in the position of the shock waves can be noticed but no reference solution exists for this case. We also observe a strong vortex under the cylinder which is much weaker in [8]. This vortex seems to be associated with a Kelvin-Helmholtz instability originating at the contact discontinuity present below the cylinder. In addition, when considering the final position of the center of mass of the cylinder, we observe a fast grid convergence of the method. The position we obtain on a 400 × 80-grid is comparable to that obtained on a 1,600 × 320-grid in [8]. With increasing resolution on grids 400 × 80, 800 × 160 and 1, 600 × 320 [8] gives t = 0.14s

0.2 0.15 y

0.1

0.05 0

0

0.25

0.5 x

0.75

1

0.75

1

t = 0.255s

0.2

y

0.15 0.1

0.05 0

0

0.25

0.5 x

Fig. 1 60 contours of fluid pressure from 0 to 28 at different times, x = y = 6.25 × 10−4

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positions (0.659, 0.132), (0.649, 0.145) and (0.641, 0.147); on the same grids, we obtain (0.64375, 0.1463), (0.64278, 0.1471) and (0.64253, 0.1471).

References 1. Aquelet, N., Souli, M., Olovsson, L.: Euler-lagrange coupling with damping effects: Application to slamming problems. Comput. Methods Appl. Mech. Eng. 195(1–3), 110–132 (2006) 2. Arienti, R., Hung, P., Morano, E., Shepherd, J.E.: A level set approach to Eulerian-Lagrangian coupling. J. Comput. Phys. 185, 213–251 (2003) 3. Daru, V., Tenaud, C.: High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193(2), 563–594 (2004) 4. Donea, J., Giuliani, S., Halleux, J.P.: An arbitrary Lagragian Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33, 689–723 (1982) 5. Dragojlovic, Z., Najmabadi, F., Day, M.: An embedded boundary method for viscous, conducting compressible flow. J. Comput. Phys. 216(1), 37–51 (2006) 6. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yusof, J.: Combined immmersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161(1), 35–60 (2000) 7. Falcovitz, J., Alfandary, G., Hanoch, G.: A two-dimensional conservation laws scheme for compressible flows with moving boundaries. J. Comput. Phys. 138, 83–102 (1997) 8. Hu, X.Y., Khoo, B.C., Adams, N.A., Huang, F.L.: A conservative interface method for compressible flows. J. Comput. Phys. 219(2), 553–578 (2006) 9. Miller, G.H., Colella, P.: A conservative three-dimensional Eulerian method for coupled solidfluid shock capturing. J. Comput. Phys. 183(1), 26–82 (2002) 10. Mohd-Yusof, J.: Combined immersed-boundary/b-spline methods for simulation of flow in complex geometries. CTR Annual Research Briefs, Center for Turbulence Research, NASA Ames/Stanford University (1997) 11. Monasse, L., Mariotti, C.: An energy-preserving discrete element method for elastodynamics. (submitted to ESAIM journal “Mathematical Modelling and Numerical Analysis”) 12. De Palma, P.M., de Tullio, D., Pascazio, G., Napolitano, M.: An immersedboundary method for compressible viscous flows. Comp. Fluid 35(7), 693–702 (2006) 13. Pember, R.B., Bell, J.B., Colella, P., Crutchfield, W.Y., Welcome, M.L.: An adaptive Cartesian grid method for unsteady compressible flow in irregular regions. J. Comput. Phys. 120, 278–304 (1995) 14. Peskin, C.S.: Numerical analysis of blood flow in the heart. J. Comput. Phys. 25, 220–252 (1977)

A Preconditioned Euler Flow Solver for Simulation of Helicopter Rotor Flow in Hover Kazem Hejranfar and Ramin Kamali Moghadam

Abstract In the present study, a preconditioned Euler flow solver is developed to accurately and efficiently compute the inviscid flowfield around hovering helicopter rotor. The preconditioning method proposed by Eriksson is applied. The three-dimensional preconditioned Euler equations written in a rotating coordinate frame are solved by using a cell-centered finite volume Roe’s method on unstructured meshes. High-order accuracy is achieved via the reconstruction of flow variables using the MUSCL interpolation technique. Calculations are carried out for an isolated rotor in hover for different conditions and the computed surface pressure distributions are compared with the experimental data. The study indicates that preconditioning increases the convergence rate of solution and improves the accuracy of the results especially at inboard sections of the blades.

1 Introduction The accurate numerical simulation of the helicopter rotor flow in hover or forward flight leads to an accurate calculation of rotor blade aerodynamic loads. The flowfield around a rotor is difficult to model because of the inherently dynamic and multidisciplinary nature of the problem [4–6, 10]. Numerical flow solvers have to be able to accurately solve a wide range of flow conditions from low speed incompressible flow near the root region of the blades to compressible high subsonic/transonic flow at the tip region. The compressible system of equations for analyzing low Mach number flows using time marching algorithms does not provide accurate results and draws low convergence rate due to the large disparity of the acoustic wave speed and the wave convected at the fluid speed. In such problems which have the regions that the compressibility is important, using the incompressible equations is restrictive. In such cases, the compressible equations should be used and the convergence

K. Hejranfar (B) Department of Aerospace Engineering, Sharif University of Technology, Tehran, Iran e-mail: [email protected]


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difficulty arising from the low speed flow can be addressed by using a preconditioning technique. The preconditioning technique basically multiplies a preconditioning matrix to the time derivative term of the governing equations. Preconditioning changes the eigenvalues of the system of compressible equations and reduces the disparity in the wave speeds and prevents inaccurate and non-physical results and resolves low convergence rate of the solution of low Mach number flows. Removal of stiffness, behaving of the system of equations as a scalar equation and decupling of the governing equations are some other advantages of preconditioning techniques [7]. Over the past decade, several preconditioning schemes have been proposed and examined by many researchers to compute compressible flow at low Mach numbers [2, 3, 11, 13, 14]. A number of Euler/Navier-stokes flow solvers have also been developed to simulate helicopter rotor flows [8]. The main objective of the present work is to develop a more economical Euler flow solver by implementing a preconditioning scheme for accurate and efficient computation of the flow around a helicopter rotor in hover. Herein, the preconditioning scheme proposed by Erikson [3] is implemented. The numerical method used is a cell-centered finite volume scheme that is based on the Roe’s flux-difference splitting method [9] on unstructured meshes. For a high-order scheme, the estimation of the flow variables at each cell face is achieved by the MUSCL formulation. The results using the present preconditioned Euler flow solver are compared with the experimental results and the effect of preconditioning on the performance of the solution is investigated.

2 Governing Equations The Euler equations are formulated in a rotating coordinate frame (x, y, z) attached to the rotor blades in terms of absolute-flow velocities. The inertial coordinates (x , y , z ) are taken to coincide with (x, y, z) at an instant in time t. The computational domain consists of unstructured tetrahedral cells (see Fig. 1). The Euler equations employing preconditioning technique may be written in conservative form as −1

c ∂ F ∂G ∂ H ∂Q + + + = S ∂t ∂x ∂y ∂z

(1)

c = [ρ ρu ρv ρw E]T is the conservative solution vector, S = [0 ρv where Q − ρu 0 0]T is the source term due to the centrifugal force of rotation of the G and H are the inviscid flux vectors in the rotating frame, is the blades and F, preconditioning matrix, is the angular velocity in the z direction relative to the inertial frame (x , y , z ) and V = (u, v, w) are the absolute velocity components relative to the inertial frame (x , y , z ). The inviscid flux vectors are expressed based ˜ . nˆ = Vn − Vñ that is the normal relative velocity in which on Unr = (V − V) V˜ = (−y, x) is the grid velocity vector.

Preconditioned Euler Flow Solver

481 10 Without Precondtioning With Precondtioning

λmax/λmin

8

6

4

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Mach number

Fig. 1 Blade surface and computational grid (left) and comparison of condition number for preconditioned and non-preconditioned systems in the rotating frame (right)

p = Equation (1) can also be written in terms of the primitive variables, Q T [ρ u v w p] as follows: p ∂Q ¯ xp ∂ Q p + ¯ A¯ yp ∂ Q p + ¯ A¯ zp ∂ Q p = S + ¯ A ∂t ∂x ∂y ∂z

(2)

¯ xp , A¯ yp and A ¯ zp are the Jacobian matrices of the inviscid flux vectors where A p . The preconditioning F, G and H , respectively, based on the primitive variables Q ¯ matrix was proposed by Eriksson [3] as follows: ⎡

1 0 0 0 − 1−a c2 ⎢0 1 0 0 0 ⎢ ¯ = ⎢0 0 1 0 0 ¯ −1 M ¯ = M ⎢ ⎣0 0 0 1 0 0000 a

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(3)

¯ = ∂Q c /∂ Q where M

p . The optimum value for the under-relaxation parameter a is a = min 1, Mr2 and Mr is the relative local Mach number. The calculation of a using this relation guaranties the modified acoustic waves propagate at nearly the same speed as the entropy and vorticity waves [3]. The eigenvalues of the preconditioned system of equations are given as

Unr

λ4,5 = Unr ± c λ1 = λ2 = λ3 = Unr , A (1 + a) (1 − a)2 2 Unr , Unr = c = ac2 + 2 4

(4)

482


Figure 1 indicates that through the preconditioning the eigenvalues become much more clustered as Mach number approaches zero.

3 Numerical Solution Procedure Equation (1) may be written in an integral form for an arbitrary grid cell as ∂ ∂t

V

Qc d V +

∂V

F(Qc ) . nˆ d S =

V

S(Qc ) d V

(5)

where V is the cell volume, nˆ dS is a vector element of surface area with outward unit normal vector n( ˆ nˆ x , nˆ y , nˆ z ). A discretization form of the governing equations can be written as d Fk . d Sk = c Sj (Q j V j ) + c dt 4

(6)

k=1

The numerical flux of the inviscid terms across each cell face k using preconditioned Roe’s flux-difference splitting can be written as k = F

1 2

R −1 ˜ ˜ Q L ) + F( ˜ F( Q ) − Q A c c c

k

(7)

Here, A˜ is the Roe-averaged flux Jacobian matrix (A = ∂F/∂Q) and Q L and Q R are the state variables to the left and right of the interface k, ( ) = ( ) R − ( ) L , and the superscript ∼ denotes Roe-averaged quantities. High-order accuracy is achieved via the reconstruction of flow variables using the MUSCL interpolation technique [12].

4 Boundary Conditions At the blade surface, the flow tangency condition is used for inviscid flows. The treatment of the far-field boundary condition is based on the one-dimensional Riemann invariants normal to the far-field boundary.

5 Results and Discussion To show the efficiency and accuracy of the present preconditioned Euler flow solver, the inviscid flowfield is computed for an isolated rotor in hover. This test case was experimentally studied by Caradona and Tung [1]. The experimental model consists of a two-bladed rigid rotor with rectangular planform blades with no twist or taper. The blades are made of NACA 0012 airfoil sections with an aspect ratio of 6.

Preconditioned Euler Flow Solver

483

Calculations are performed for the operating conditions of subsonic and transonic tip Mach numbers, Mtip = 0.44 and 0.877, for the collective pitch angle θc = 8◦ . The present results are compared with the experimental data and the effect of preconditioning on the performance of the solution is investigated. Figure 1 shows the blade surface and computational grid used. Figure 2 compares the computed surface pressure coefficient distributions with the experimental data at different spanwise locations for Mtip = 0.44 and 0.877. The results for the preconditioned and non-preconditioned systems are nearly the same and they are in good agreement with those of experiment. It is obvious that the transonic region near the tip of the blades for the Mtip = 0.87 test case is accurately predicted. For both the cases, the second-order solution provides more accurate results than the first-order one. Figure 3 gives the effect of preconditioning on the convergence rate of the solution for both the subsonic and transonic test cases. It is demonstrated that

Fig. 2 Comparison of surface pressure coefficient distribution for Mtip = 0.44 (left) and Mtip = 0.877 (right) 10–3 10

10–5

10–5

–6

10–6

10

10–7 –8

10–7 10–8

10

10–9 10

Without Preconditioning With Preconditioning

10–4

Error

Error

10–3

Without Preconditioning With Preconditioning

–4

10–9

Mtip = 0.44

–10

0

5000

10000 Iteration

15000

10–10

Mtip = 0.877 0

5000

10000 Iteration

15000

Fig. 3 Effect of preconditioning on convergence rate of the solution for Mtip = 0.44 (left) and Mtip = 0.877 (right)

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preconditioning greatly enhances the performance of the solution for the flow around the rotor for both the cases. Note that a large region of the flowfield around the rotor has the characteristics of a low speed flow, and therefore, preconditioning becomes more efficient for simulating this class of the flow. The study indicates that the effect of preconditioning in improving the convergence rate of the solution for the subsonic case is more pronounced than the transonic one. Acknowledgements The author would like to thank Sharif University of Technology for the financial support of this study.

References 1. Caradonna, F.X., Tung, C.: Experimental and analytical studies of a model helicopter rotor in hover. Vertica 5, 149–161 (1981) 2. Choi, Y.H., Merkle, C.L.: The application of preconditioning in viscous flows. J. Comput. Phys. 105, 207–223 (1993) 3. Eriksson, L.E.: A preconditioned Navier-Stokes solver for low Mach number flows. In: Désidéri, J.-A., Le Tallec, P., Oñate, E., Périaux, J., Stein, E. (eds.) Proceedings of the Third ECCOMAS Computational Fluid Dynamics Conference, pp. 199–205. John Wiley & Sons, Paris, France (1996) 4. Hall, C.M., Long, L.N.: High-order accurate simulations of wake and tip vortex flowfields. AHS 55th Annual Forum, Montreal, Canada (1999) 5. Hariharan, N., Sankar, L.N.: Higher order numerical simulation of rotor flow field. AHS Forum and Technology Display, Washington, DC (1994) 6. Hariharan, N., Sankar, L.N.: First-principles based high order methodologies for rotorcraft flowfield studies. AHS 55th Annual Forum, Montreal, Canada (1999) 7. Lee, D.: Design criteria for local Euler preconditioning. J. Comput. Phys. 144, 423–459 (1998) 8. Potsdam M.A., Sankaran V., Pandya S.A.: Unsteady low mach preconditioning with application to rotorcraft flows. Proceedings of the 18th AIAA CFD Conference, Miami, FL (2007) 9. Roe, P.L.: Characteristic-based schemes for the Euler equations. Annu. Rev. Fluid Mech. 18, 337–365 (1986) 10. Tang, L., Baeder, J.D.: Improved Euler simulation of hovering rotor tip vortices with validation. AHS 55th Annual Forum, Montreal, Canada (1999) 11. Turkel, E.: Preconditioned methods for solving the incompressible and low-speed compressible equations. J. Comput. Phys. 72, 277–298 (1987) 12. van Leer, B.: Towards the ultimate conservative difference scheme, V: a second order sequel to Godunov’s method. J. Comput. Phys. 32, 101–136 (1979) 13. van Leer, B., Lee, W.T., Roe P.L.: Characteristic Time-Stepping or Local Preconditioning of the Euler Equations. AIAA Paper 92-1552 (1992) 14. Weiss, J.M., Smith W.A.: Preconditioning applied to variable and constant density flows. AIAA J. 33, 2050–2057 (1995)

A New Type of Gas-Kinetic Upwind Euler/N-S Solvers Lei Tang

Abstract A non-relaxation gas-kinetic scheme is developed, which recovers the accuracy of the BGK scheme with higher computational efficiency and without relaxation parameter. The extension of this approach to the Navier – Stokes equations further provides a solution for development of Discontinuous Galerkin method for the Navier – Stokes equations.

1 Introduction While the development of the gas-kinetic CFD solvers starts from as early as 1960s, the first gas-kinetic upwind scheme is probably the Equilibrium Flux method (EFM) in [5] or the Kinetic Flux-Vector Splitting (KFVS) method in [1, 3]. Although it has some unique features, a KFVS scheme is found more diffusive than Roe’s approximate Riemann solver in [6]. To improve the accuracy over a KFVS scheme, the so-called Bhatnagar–Gross–Krook (BGK) method is developed in [4, 9], making a relaxation between the KFVS scheme and another less robust gas-kinetic scheme, KFVS_u 0 . However, the relaxation approach involves a free parameter similar to the artificial viscosity used in the central schemes. As a result, the benefits of an upwind shock-capturing scheme are not fully materialized. So, in this chapter, we will explore a non-relaxation approach to recover the accuracy of the BGK scheme.

2 Problem Statement For the Euler equations, the Kinetic Flux-Vector Splitting (KFVS) method in [3] splits the flux vector, F, according to the sign of the particle velocity u,

L. Tang (B) D&P LLC, Phoenix, AZ 85016, USA e-mail: [email protected]


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ρ

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+

F=F +F

−

=

ψug L dΞ + u>0

ψug R dΞ u0

u0 ψg L dΞ + u0 u 2 u0 u 3 g L dΞ + u0 gx L dΞ 5 u0 u 0 is introduced to avoid the denominator to become 0 and we take ε = 10−8 . The properties of this procedure will be discussed elsewhere for brevity. It has been shown that this procedure is quadrature- free, parameter-free and non-oscillatory (QPN). It is therefore called the QPN limiting procedure in the present paper. 3. The right states in terms of the characteristic variables are computed in a similar way. Finally, the left and right states, U L (xm,ϕ ), U R (xm,ϕ ), used in Eq.(2) are deduced from the characteristic variables.

3 Numerical Tests The first test case is to solve a 1D nonlinear Burgers equations using 3rd order FV scheme using the QPN limiting. The governing equation and the initial/boundary conditions are u t + (u 2 /2)x = 0, 0 x < 2, t > 0 u(x, 0) = 0.5 + sin(π x), u(0, t) = u(2, t) At t = 0.5/π , the solution is still smooth. An accuracy test on uniform grids is performed to find the order of convergence of the scheme with and without QPN limiting. The numerical results are presented in Fig. 1(a). To show the shock-capturing capability of the proposed scheme, the numerical results are presented at a series of time as depicted in Fig. 1(b) and no oscillation occurs in the vicinity of shock. The second test case is the double mach reflection problem [8]. The computational domain for this problem is [0, 4] × [0, 1] and a wall lies on the bottom of the computational domain starting from x = 1/6. A Mach 10 shock that makes a 60◦ angle with the x-axis moves from left to right. This test case is solved respectively by the 3rd order FV scheme using QPN limiting with 90,652 grids. The numerical results at T=0.21 are shown in Fig. 2 in terms of the density contours. It is clear that QPN limiting can capture the shock waves and contact discontinuities sharply in Fig. 3(a). And no oscillation occurs as shown in Fig. 3(b).

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4 Conclusions This paper presents a family of high order FV schemes applicable on general shaped control volumes. Similar to the usual k-exact reconstruction, the reconstruction is performed only once on each control volume. Multiple reconstructions used for the weighted averaging procedure are constructed through the modification of the reconstructions on adjacent cells by using a very simple and efficient secondary reconstruction procedure. A WBAP limiter in terms of characteristic variables is proposed. From the test cases it shows that the numerical schemes of the present paper can predict the smooth fields with uniformly high order accuracy and can capture the shock waves and contact discontinuities in high resolution.

References 1. Barth, T.J., Frederichson, P.O.: Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. AIAA 90-0013 (1990) 2. Barth, T.J., Jespersen, D.: The design and application of upwind schemes on unstructured meshes. Proceedings of the 27th AIAA Aerospace Sciences Meeting, Reno, NV, Paper AIAA 89-0366 (1989) 3. Choi, H., Liu, J.G.: The reconstruction of upwind fluxes for conservation laws: its behavior in dynamic and steady state calculations. J. Comput. Phys. 144, 237–256 (1998) 4. Hu, C., Shu, C-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999) 5. Jiang, G.S., Shu, C.W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996) 6. Ollivier-Gooch, C.F.: Quasi-ENO schemes for unstructured meshes based on unlimited datadependent least-square reconstruction. J. Comput. Phys. 133, 6–17 (1997) 7. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988) 8. Woodward, P., Colella, P.: Numerical simulations of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54, 115–173 (1984)

Multiblock Computations of Complex Turbomachinery Flows Using Residual-Based Compact Schemes Bertrand Michel, Paola Cinnella, and Alain Lerat

Abstract A robust implementation of residual-based compact schemes of secondand third-order accuracy on structured multiblock grids of industrial use is proposed. Applications to complex transonic unsteady flows in turbomachinery show the advantages of using such schemes in terms of increased accuracy and better insight into the flow physics.

1 Introduction Turbomachinery flows are three-dimensional, unsteady, and often transitional. In the case of modern highly loaded aircraft engines turbomachinery, they may also exhibit moving shock waves, shock/boundary layer interaction and boundary layer separation. Flow unsteadiness is due to relative motion of the rotors with respect to the stators, to intrinsic flow instabilities like the shedding of vortices behind blunt trailing edges of the blades and shock buffeting phenomena, to random unsteadiness in the freestream, and to interactions of endwall flows (secondary and leakage flows) with the subsequent blade row. In addition to physical complexities, turbomachinery flows are also characterized by complicated geometries, involving twisted blades, tip clearance, rotor/stator relative movement, etc. In an industrial context, the computational domain is often discretized by relatively coarse grids decomposed into several subdomains. In these conditions, the accuracy of classical numerical schemes, which is typically nominally second-order, may be decreased to first-order only. Nevertheless, numerical errors play an important role in the computation of turbomachinery performance. For instance, dissipative schemes and coarse meshes do not provide a correct description of vortex streets, which represent a substantial component of the blade profile losses. This work is a step toward the development of a new generation of simulation tools for in-detail turbomachinery flow analyses. Recently developed Residual-Based Compact (RBC) numerical schemes of 2nd and 3rd-order accuracy [5, 6] are implemented within the structured multiblock finite B. Michel (B) Département DSNA ONERA, 92320 Châtillon, France e-mail: [email protected]


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volume solver elsA [1] developed by the DSNA Department of ONERA. The performance of RBC schemes is investigated for unsteady flows in increasingly complex turbomachinery geometries.

2 Residual-Based Compact Formulation Turbomachinery flows are modelled by the Reynolds-Average Navier-Stokes equations completed by a transport equation model. For the following computations, we retain the Spalart-Allmaras model, among different choices available in elsA [1]. For simplicity, here the scheme is described for the 2D Euler equations: wt + f x + g y = 0,

(1)

where w is the conservative variable vector, f = f (w) and g = g(w) are flux functions in the x and y directions of a Cartesian frame, and the subscripts denote partial differenciation. Equation (1) is approximated through a linear multistep method of the form: q 1 αl (Δl w)n+1 + R(w n+1 ) = 0 Δt

(2)

l=1

where Δwn+1 = w n+1 − wn , Δt is the time step, αl = 1/l, and R is a spatial approximation operator. For q = 1, Eq. (2) gives the 1st-order backward Euler scheme, and for q = 2, it is the 2nd-order Gear scheme. Both methods are A-stable. The updated solution at time n + 1 is computed by solving the nonlinear Eq. (2). This is done using Newton subiterations along with a defect-correction technique [1]. The spatial operator R is constructed by using a RBC approach [5, 6]. Given a uniform Cartesian mesh, the 3 × 3 point RBC scheme reads: q δ12 δ22 1 δ1 h 1 δ2 h 2 n+1 αl (Δl w) + = 0, (3) I +θ + I +θ 6 6 Δt δx δy j,k l=1

where I is the identity operator, θ ∈ [0, 1] is a real parameter, δ1 (•) j+ 1 ,k := 2 ((•) j+1,k − (•) j,k ) is the difference operator over one cell in the x direction, δ2 is the analogous operator in the y direction, and h 1 and h 2 are numerical fluxes in the x and y directions. For instance: 3 (h 1 ) j+ 1 ,k = 2

δ2 I +θ 2 6

δx μ1 f − Φ1 r˜1 2

4 ,

(4)

j+ 12 ,k

where μ1 (•) j+ 1 ,k := 12 ((•) j+1,k − (•) j,k ) is the average operator over one cell in 2 the x direction, Φ1 a dissipation matrix, and r˜1 a 2 × 3 point approximation of the residual –LHS of Eq. (1)– on the cell face. An analogous writing is used for h 2 . The

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BLOCK 1

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Fig. 1 Exchanges between subdomains for an ideal configuration

preceding scheme is 1st-order accurate during the Newton sub-iteration process, and high-order accurate when the discrete unsteady residual tends toward zero. Precisely, the RBC numerical approximation is 2nd-order for θ = 0 and 3rd-order for θ = 1. In the following, the 2nd-order scheme is referred-to as RBC2, and the 3rd-order one as RBC3. The schemes are extended to general grids using a cell-centered finite volume formulation. For RBC3, the weighted formulation of [7] is adopted to preserve high accuracy on deformed meshes. The weighted RBC3 scheme is referred to as RBC3i. Preliminary tests showed that the weighted formulation prevents the appearence of spurious high-frequency oscillations on very distorted meshes. The computational stencil of RBC schemes involves corner cells of the subdomains. For instance, let us consider a rectangular domain split into 4 subdomains (Fig. 1). Once one subdomain (say 1) is isolated from the rest, the computation can be carried out independently from neighbouring subdomains, if the ghost-cell value in the corner on the bottom-right is provided from 4. Conversely, subdomain 4 needs information on the corner value from 1. This approach is satisfactory for simple configurations. For complex mesh topologies corner cells may overlap more than one subdomain, and the preceding procedure is ill-posed. For the sake of generality, residuals at corner cells are computed by means of a directional approximation: precisely, a simple centered second-order scheme or a 5-point Directional Non-Compact scheme of third-order accuracy [3] (noted DNC3 in the following) are used, along with a directional Jameson-type artificial viscosity. Numerical tests show that the proposed implementation improves robustness, reduces computational costs and preserves accuracy compared to the exact RBC treatment.

3 Results The RBC2 and RBC3i schemes are applied to the computation of transonic unsteady flows in turbomachinery. Results are compared with those of Jameson’s scheme and of DNC3 scheme implemented within the same code.

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First, we simulate unsteady transonic flow through the VKI LS-59 turbine cascade. This is a high-loaded rotor blade with a thick, rounded trailing edge originally designed for near-sonic exit flow conditions, and extensively tested in various European wind tunnels [4]. Experiments are available in a wide range of conditions, and Schlieren photographs clearly indicate the existence of vortex shedding downstream of the blade blunt trailing edge, which is responsible for an appreciable fraction of profile losses. The flow conditions considered for this study correspond to a unit exit isentropic Mach number and a Reynolds number of 7.44×105 based on the chord length and mean outlet velocity. The inlet flow angle is equal to 30. Unsteady two-dimensional computations are carried out using a C-grid of 384 × 32 cells. The unsteady computations are initialized via a preliminary steady run. The nondimensional time step is selected in order to get approximately 30 time steps per shedding cycle. Snapshots of the instantaneous pressure contours and entropy field around the blade are provided in Fig. 2 for the RBC2, RBC3i, Jameson and DNC3 schemes. Only the RBC3i scheme predicts vortex shedding. Fig. 3 provides the Fourier spectra for the four schemes. The RBC3i scheme returns a rich frequency spectrum with a dominant frequency corresponding to a well-defined peak in amplitude and a large band of harmonics. The computed Strouhal numbers, based on the main frequency, the isentropic outlet velocity and the thickness of the trailing edge (equal to 0.06 the axial chord for this blade) are of the order of 10−2 for all the schemes, except RBC3i that returns a value of 0.206, close to the experimentally measured range [0.2, 0.4]. Then, the RBC schemes are used to compute the BRITE HP transonic turbine stage, experimentally tested in the compression tube facility CT3 of the Von Karman Institute [2], at pressure ratio 5.11. Quasi-3D unsteady computations are performed on a radial portion of the turbine discretized by means of 5 mesh points in the

Fig. 2 VKI LS-59 turbine. Snapshots of the unsteady entropy field and isobars for four schemes. Left to right and top to bottom: RBC2, RBCi, Jameson, DNC3

Multiblock Computations of Complex Turbomachinery

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Fig. 3 VKI LS-59 turbine. Fourier spectra of the unsteady axial force on the blade

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Fig. 4 Instantaneous entropy contours and isobars for transonic flow through BRITE HP turbine. Left: Jameson, right: RBCi

radial direction. The computational grid contains approximately 150,000 cells and is composed by 8 blocks: both the rotor and the stator are discretized by an O-shaped grid around the blades and H-shaped blocks for inlet, outlet, and inter-blade regions. Unsteady computations are initialized with steady results obtained by imposing a mixing-plane inter-stage condition. Chorochronic boundary conditions allow simulating just one blade per stage. Figure 4 shows the entropy contours and isobar lines on the turbine middle plane for the schemes under investigation. RBCi sharply captures the shock waves and von Karman streets, which in turn are smoothed out by Jameson’s scheme. RBC2 and DNC3 display an intermediate behavior. Finally, because of higher losses, Jameson’s scheme predicts a lower mean mass flow than DNC3, RBC2 and RBC3i for the prescribed pressure ratio: the normalized mass flows are 0.888, 0,897, 0.908, and 0.913, respectively.

4 Discussion and Final Remarks Computational results for complex unsteady turbine flows demonstrate the superior performance of the RBC schemes with respect to more standard approaches on coarse computational grids. Specifically, the RBCi scheme of third-order accuracy

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captures fine flow details on coarse meshes of industrial use. For steady computations, its computational cost per iteration and per point is about 2 times that of Jameson’s scheme. For unsteady computations, the over-cost for RBCi is multiplied by about 1.5–2 because of the greater number of Newton subiterations required. On the other hand, capturing the same physical phenomena using lower-order schemes requires much finer meshes: numerical tests (not included for brevity) show that a 512 × 64 cell mesh in conjunction with a smaller time step and a cutoff of the eddy viscosity are required to capture vortex shedding using a Jameson-type scheme. For very complex problems, this implies a significant amount of trial and error to capture the correct flow physics, with subsequent increase of the global turn-around time for a simulation. In summary, the RBC3i scheme provides a more detailed solution with a reduced need for adjustment of simulation parameters, and opens the way to more reliable simulations of complex turbomachinery flows.

References 1. http://elsa.onera.fr/ 2. Denos, R. et al.: Investigation of the unsteady Rotor Aerodinamics in a Transonic Turbine Stage. ASME Paper 2000-GT-435 (2000) 3. Huang, Y., Cinnella, P., Lerat, A.: A third-order accurate centered scheme for turbulent compressible flow calculations in aerodynamics. In: Morton, B. (ed.) Numerical Methods in Fluid Dynamics, VI, p. 355. Will Print, Oxford (1998) 4. Kiock, R., Lehtaus, F., Baines, N.C., Sieverding, C.H.: The transonic flow through a plane turbine cascade as measured in four european wind tunnels. J. Eng. Gas Tur. Power 108, 810– 819 (1996) 5. Lerat, A., Corre, C.: A residual-based compact scheme for the compressible Navier-Stokes equations. J. Comput. Phys. 170, 642–675 (2001) 6. Lerat, A., Corre, C.: Residual-based compact schemes for multidimensional hyperbolic systems of conservation laws. Comp. Fluids 31, 639–661 (2002) 7. Lerat, A., Corre, C., Hanss, G.: Efficient high-order schemes on non-uniform meshes for multiD compressible flows. In: Caughey, D., Hafez, M. (eds.) Frontiers of Computational Fluid Dynamics, p. 89. World Scientific, Singapore (2002)

A Local Resolution Refinement Algorithm Using Gauss–Lobatto Quadrature J.S. Shang and P.G. Huang

Abstract The polynomial local resolution refinement has been successfully applied to the counter-flow hydrogen combustion simulation. The solution continuity across the staggered grid interface is reinforced by an efficient finite-difference reconstruction approximation using roots of the Gauss–Lobatto polynomial. Meanwhile, the spectral numerical resolution is verified by the L2 norm projection formulation.

1 Introduction The molecular diffusion, turbulent mixing, and reaction rates of chemical kinetics control the fine structure of a flame. All these processes are occurred on the atomic and molecular levels and usually beneath the Kolmgorov microscales [7]. Therefore, the approach by using the large eddy simulation (LES) may not be adequate to achieve the desired objective and an alternative is urgently needed. Using the formal order of accuracy for truncation error exclusively to select an algorithm for numerical simulation over an isolated high-gradient domain is an over simplification. The desired feature of a numerical scheme needs be derived from the intermediate wave number to maintain a lower level of dispersive and dissipative error than conventional numerical schemes. In addition, the polynomial refinement appears to be very attractive because it is a basic one-dimensional operator and equally applicable to temporal and spatial dimension [6]. The Gauss quadrature employs unequally spaced roots or base points that are diametrically different from the Newton–Cotes formulation. In application, the variables between two temporal or spatial cells can be approximated by any eight known orthogonal polynomials from Legendre to Meixner–Polluckzek, but the former is the mostly widely used because of the accumulative knowledge of this function [2]. The performance and unique features of polynomial refinement in a steep gradient

J.S. Shang Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH, USA e-mail: [email protected]


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domain have recently been reasonably established for further research and pathfinding applications [3, 11]. Applying the polynomial refinement to a hyperbolic conservative law requires only balancing the outward normal flux vector component on the interface boundary between global grids. Therefore the solution is permitted to be discontinuous at the interface to satisfy the domain of dependence condition. The approximate Riemann approach is well-posed [8]. However when solve the diffusion equation, the interaction of data between global blocks must be maintained for functional continuity across the interface and this issue has been known at the very beginning of the polynomial refinement algorithm development [6]. A wide range of procedures have been investigated from the penalty method [1] to the more recent work on the recovery polynomial basis [10]. The essential idea is rest on an effective coupling of the discontinuous solutions across the interface of global grids which permits a very large class of solving schemes to be utilized [4]. This issue becomes a focused area of the present studied. The high gradient region of a flame front is physically different from the hydrodynamics shock which is formed by just a few collisions among gas molecules. On the other hand, the common feature of the flame front is reflecting by a steep gradient region confined by the diffusion and chemical reaction phenomena which are the inherent process of combustion [7]. The detailed flame structure is still beyond our understanding because the time and length dimensions are smaller than the Kolmogorov turbulent scales. A sharp demarcation of a flame structure requires a high-resolution description. The present investigation attempts to achieve a better understanding of this complex and fine-scale physic-chemical phenomenon through high-resolution numerical simulations.

2 Analysis In applying the polynomial refinement, the first step is to approximate a function in an arbitrary integral interval by the Gauss quadrature using Legendre polynomials through an independent variable transformation. The Gauss-Lobatto formulation yields an approximation to the function, F(x), and the accuracy is uniquely determined by the degrees of the polynomial. 3 n 4 n H F n+1 (ξ ) F(x) = Pn (x) + Rn (x) = L i (x)F(xi ) + (x − xi ) (1) (n + 1)! i=1

i=1

The weight (Cardinal) function of the approximation is obtained as the product of the divided-difference approximation [2]; n

L i (x) = '

j=0 j=i

(x − x j ) (xi − x j )

(2)

The derivative of the approximation function within the integral interval can be computed by simply differentiating the Cardinal function L i (x).

Local Resolution Refinement Algorithm Using Gauss–Lobatto Quadrature l

l q=1,q=i dF(xi ) dL i (x) = f (xn ) = dx dx n i=1

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(x p − x t ) F(x n ).

(xi − x m ) (3)

It is immediately recognized that the above formulation provides the recursive relationship for the first-order derivative computation through the Gauss quadrature and can be expressed by an n × n matrix as the follows; ⎡

a1,1 (x 1 ) ⎢ a2,1 (x 1 ) dF(xi ) ⎢ =⎢ .. ⎢ dx ⎣ an−1,1 (x 1 ) an,1 (x1 )

a1,2 (x2 ) a2,2 (x2 ) .. an−1,2 (x2 ) an,2 (x 2 )

a1,3 (x3 ) a2,3 (x3 ) .. an−1,3 (x3 ) an,2 (x 3 )

.. .. .. .. ..

a1,n−1 (x n−1 ) a2,n−1 (x n−1 ) .. an−1,n−1 (x n−1 ) an,n−1 (xn−1 )

⎤⎡ ⎤ a1,n (xn ) F(x1 ) ⎢ ⎥ a2,n (xn ) ⎥ ⎥ ⎢ F(x2 ) ⎥ ⎥⎢ ⎥ (4) .. . ⎥⎢ ⎥ an−1,n (xn ) ⎦ ⎣ F(xn−1 ) ⎦ F(x n ) an,n (x n )

This approximation is restricted to generate the first derivative of the approximated function, because all hypogeometric differential equations that associated with the orthogonal polynomials are second order. When attempts to generate the second derivative from the weighed function, the eigenvalue structure often leads to oscillatory behavior [3, 9]. For this reason, the second spatial derivative in the present anlysis is obtained by consecutive approximations using the calculated first derivative. A unique feature of the local polynomial refinement approach is that it’s independent from the global mesh system. There is no need to reconstruct the overall grid system for local refinement to capture the fine-structure features, but by just increasing the degrees of polynomials within the grid block. The numerical procedure is equally applicable to the temporal advancement of a dynamic problem [3, 9]. In some cases, it may even be possible to evaluate integrals in which the integrand has discontinuity within the integral interval. The singularity can be relegated to the weighting function [2, 9]. In order to apply the approximate Riemann formulation on the control surface, the Gauss–Lobatto quadrature offers a distinguished advantage. By this formulation, the integral limits of the Gauss quadrature become two of the root points [2, 3, 9] and the quadrature now involves the weighted sum of n + 1 functional values. The present analysis is focused on the solution to the canonical equation for combustion by developing the polynomial refinement procedure for numerical simulation. Thus, the most detailed study is concentrated on the species conservation equation in the following. When the conservation law is expressed in mass concentration of chemical species, ci , it becomes; ρ

∂ci + u · ∇ci ∂t

− ∇ · (ρ Di j ∇ci ) =

dwi , dt

(5)

where the species velocity, u i , is given by the Fick’s first law for molecular diffusion, which is calculated as (ρu)i = −ρ Dij ∇ci . The rate of species generation and

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depletion in combustion on the right-hand-side of the equation is calculated from the law of mass action [5]. In order to highlight the viability of using polynomial refinement for a thin flame, the species concentration equation is further reduced to its limiting one-dimensional form on the axis of symmetry. The asymptotic form derived from a cylindrical polar coordinate by applying the L’hospital rule at the limit of vanishing radius yields; ∂ 2 ci 1 ∂ρ Di j ∂ci dwi ∂ci + ux − − Di j 2 = ∂t ρ ∂x ∂x ∂x dt

(6)

The above one-dimensional, species conservation equation along the axis of symmetry is adopted as the physical-based model equation for the polynomial refinement study of a thin flame. The finite-difference approximation of Eq. (6) in discretized space is formulated according to the delta formulation, except all derivatives on the right-hand-side of the equation are evaluated by the high-resolution Gauss–Lobatto formula. The resultant finite-difference equations can be solved easily by an implicit tridiagonal scheme. n+1 α1 ci−1

+ α2 cin+1

n+1 + α3 ci+1

n ∂ 2 ci dwi ∂ci ∂ci = − + u˜ − Di j 2 dt ∂t ∂x ∂x

(7)

The right-hand-side of the above equation is evaluated by the Gauss quadrature in time and space to achieve the high resolution. The left-hand-side terms are manipulated to achieve a diagonal dominant, unconditionally stable, windward approximation. In which the convective terms are solved by the Roe’ approximate Riemann and the diffusive terms by a simple spatial central scheme [8]. The identical numerical procedure is applied both to the overall grid system of the uniform spaced blocks and the locally refinement within the block by the unevenly distributed root points of the orthogonal polynomial. For solving a multi-dimensional problem, the procedure is identical to the classic ADI scheme and has been successfully illustrated by Huang et al. [3]. The polynomial approximation of a function at the interface of multiple grid blocks is generally discontinuous [4, 9]. When the diffusion terms are dominant in the partial differential equation, the mathematical conditions of solution at the interface expands to require symmetry, coercivity, boundedness, consistency, and adjoint consistency [10]. Regardless of these complexities, the essence of physics requires information exchange between global blocks and boundary conditions form the computational domain [4, 10]. This requirement is met by a by an increasing number of root points for increasing accuracy until a desired resolution is reached [4]. dF dF = [F(xi−1,n−1 ), F(xi−1,n−2 ), F(xi−1,n−3 ), ....F(xi,1 ), F(xi,2 ), F(xi,3 ), ....] dx dx d2 F d2 F = [F(xi−1,n−1 ), F(xi−1,n−2 ), F(x i−1,n−3 ), ....F(xi,1 ), F(xi,2 ), F(xi,3 ), ....] dx2 dx2

(8) (9)

Local Resolution Refinement Algorithm Using Gauss–Lobatto Quadrature

519

The above simple yet effective reconstruction technique parallel to that of Huynh [4] is implemented to preserve functional continuity at the interface of the global grid block. The process is needed only for the derivatives at the interface. The continuity of the solutions across the global grid is achievable by this approach and the high-resolution numerical result is verified by the accuracy assessment through a comparison with the L2 norm projection approach [10].

3 Numerical Results The numerical results consist of two groups; the first group of results is devoted to preserve the high-resolution solution of the one-dimensional heat model equation using the reconstruction technique at the global grid interface. The second group results are concentrated on the detailed structure of a counter flowing hydrogen-air diffusion flame. The initial condition of the heat equation represents a dissipating hightemperature source in the middle of the computational domain [9]. A total of 50 global grid blocks with equal dimension of 0.005 is used for numerical simulations and within each grid block the degree of polynomials has an assigned range from 11 to 51. The degree of polynomial within each contiguous grid block is different depending on the numerical resolution requirement. Figure 1 depicts the first derivative in a narrow range of −0.015 < × < 0.020 across three global grid blocks. The magnitude of the derivative spans a range of ±0.7207 × 105 in the high-gradient region. The first derivative is mostly continuous across all 50 global blocks, except in the high gradient domain abridged five global blocks. The difference between the derivatives calculated by the L2 norm projection and the reconstruction approximation diminishes rapidly, and the difference reduces only slightly as the number of the root points increased. A small numerical oscillation appears in the three-point approximation when applies without the benefit of a symmetric root-points distribution across the block interface. The minimum grid spacing between root points is 7.194 × 10−5 at the block interface which is much smaller than that of the global grid block. The numerical resolution is thus also enhanced by the highly clustered root points at the integral limits of the Gauss– Lobatto formulation. The comparison of the second derivatives calculated by the reconstruction approximations and the L2 norm projection is presented in Fig. 2. The magnitude of the derivative spans a range from −0.3754 × 108 to 0.9616 × 108 in the isolated high-gradient region. The similar behavior is observed as the comparison of the calculated first derivatives. The numerical difference with the L2 norm projection is generally less than four digits accuracy at the grid-block interface. A superior performance of the five-point formulation over the three-point approximation is noted when the adjacent blocks are assigned by two polynomials of vastly different degrees. In general the difference is small, for this reason a seven-point formulation is not attempted. In short, the present reconstruction approximation abridging the

520


Fig. 1 Comparison of first spatial derivatives

global grid blocks is an effective approach for maintaining the continuity of numerical solutions across the interface of grid blocks. The advantage of computational efficiency is by a factor of 2.25 times greater than that of the L2 norm projection method for solving the one-dimensional diffusion equation. The continuous condition of the dependent variable is not directly enforced, but as a consequence of the continuous derivatives at the grid-block interface. An important concern of the present approximation for reinforcing solution continuity at the interface is the possible degradation of numerical resolution. This concern is alleviated by examining the L2 norm value of solution normalized by the consistent polynomial overlapping results across the interface. Figure 3 presents the L2 norm of 3-point and 5-point interface solutions normalized by the result of polynomial overlapping grid blocks calculations. All solutions are sampled at the identical time


521

Fig. 2 Comparison of second spatial derivatives

elapsed, after 2000 temporal steps at a t = 10−3 . The solutions using the reconstruction approximations at the interface retain the functional continuity and exhibit negligible degradation of numerical accuracy within the grid block. The magnitude of the L2 norms generally has a value on the order less than 10−12 . The maximal value of 0.5066 × 10−11 is located at the integral limits of the interface which associates with the 3-point approximation as expected. The developed multi-grid-block polynomial resolution approach is now applied to a counter-flow hydrogen-air combustion field which has an axisymmetric configuration with a radius of 25 mm and a separation distance between nozzles of 40 mm. The air is injecting at a velocity of 5 m s−1 and the nitrogen-hydrogen mixture is issuing from the opposite nozzle at a velocity of 6 m s−1 . For the combustion field, the strain rate is 275.0 s−1 . A detailed chemical kinetics model is adopted for describing the hydrogen-air combustion

522


Fig. 3 L2 norm of approximated results normalized by on overlapped polynomials computation

process. This nonequilibrium chemical reacting model consists of 13 species – H2 , O2 , H, O, OH, H2 O, HO2 , H2 O2 , N, NO, NO2 , N2 O, and N2 by 74 elementary reactions. The comparison of the hydroxide composition in molar fraction via the Gauss– Lobatto formula and the finest-mesh result (401 × 41) by the UNICORN program is depicted in Fig. 4. The high-resolution solution is achieved on a 16 global grid blocks with three different degrees of polynomial of 11, 25, and 51 derived from the knowledge of the converged result of UNICORN [5]. The highest degree polynomial (51) is applied only in the 8th grid block (17.5 ≤ r ≤ 20 mm). The immediately adjacent blocks are assigned the 25-degree and the 11-degree polynomials farther away from the high-gradient region. The agreement between results is anticipated because the shared kinematic and thermodynamic data from the UNICORN program [5]. The only meaningful conclusion can be drawn from this comparison is that the Gauss–Lobatto formulation has the desired resolution characteristic of the intermediate wave number. It offers a potentially great advantage in practical application for combustion simulation. As a final assessment of the present analysis, the flame thickness of the counterflow hydrogen-air combustion in hydroxide concentration is compared with the results from UNICORN. In Fig. 5, the predicted hydroxide profile in terms of molar fraction duplicates the result of UNICORN. The maximum deviation is less than 4.172 × 10−3 in molar fraction. Meanwhile, the finest local root spacing for resolving the flame has reached a value less 2.309 × 10−3 mm in contrast to the value of 2.5 mm for the global grid. The flame thickness determined from the radical component, OH, of the hydrogen-air combustion is slightly thicker than that


523

Fig. 4 Comparison of molar fraction of hydroxide on axis of symmetry

Fig. 5 Flame thickness of air-hydrogen counter-flow jet combustion

of the molecular component of hydrogen. In general, the flame thickness varies from chemical species to species according to the associated diffusion velocity and chemical reaction rate. Nevertheless, the flame thickness can be characterized by the thermal thickness of the flame. The counter-flow jet velocity exhibits a complex pattern which decelerates toward the flame and accelerates at the outer edge of the flame then finally reaches a free-stream stagnation point within the flame of the combustion process. The complex flame structure is clearly governed by the

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molecular diffusion and the rate of species generation and recombination. However, an in-depth analysis can be achieved only by examining all chemical species within the flame.

4 Concluding Remarks The present study has shown that a continuous solution to the diffusion equation across the global blocks using polynomial for local resolution refinement can be ensured by reconstruction approximations at the interface using the root points of contiguous grid blocks. The high resolution is retained within the coupled grid blocks as shown by the L2 norm value normalized with respect to the solutions generated through the overlapping polynomial over immediately adjacent global grid blocks. The present result reaches a good agreement with solution of an established Unicorn computer program for combustion. The most important observation is that the Gauss–Lobatto formulation has the desired characteristic for the intermediate wave number of an isolated high gradient region. The finest spatial resolution in this region is less than ten thousandth of the global mesh spacing. Additional studies of the polynomial refinement method for its consistent adaptability, numerical stability, and computational efficiency for a linear model equation are still in process. Acknowledgements The sponsorship of Dr. F. Fahroo of Air Force Office of Scientific Research is deeply appreciated. The unfailing support and fruitful exchange with Dr. V.R. Katta of Innovative Scientific Solution Inc. is sincerely acknowledged.

References 1. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982) 2. Gautsche, W.: Orthogonal Polynomials, Computational and Approximation. Oxford Science Publications, Oxford University Press, New York, NY (2004) 3. Huang, P.G., Wang, Z.J., Liu, Y.: An implicit space-time spectral difference method for discontinuity capturing using adaptive polynomials. AIAA 2005-5255, Toronto, Canada (2005) 4. Huynh, H.T.: A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods. AIAA 2007-4079 (2007) 5. Katta, V.R., Roquemore, W.M.: Calculation of multidimensional flames using large chemical kinetics. AIAA J. 46, 1640–1650 (2008) 6. Kopriva, D.A.: A conservative stagger-grid Chebyshev multidomain method for compressible flows, II semi-structured method. J. Comp. Phys. 128, 475 (1996) 7. Pitsch, H.: Large-eddy simulation of turbulent combustion. Annu. Rev. Fluid Mech. 38, 435–482 (2006) 8. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference scheme. J. Comp. Phys. 43, 351–372 (1981)


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9. Shang, J.S.: A high-resolution method using adaptive polynomials for local refinement. AIAA 2010-0539, Orland, FL (2010) 10. van Raalte, M., van Leer, B.: Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion. Comm. Comp. Phys. 5(2–4):683–693 (2009) 11. Wang, Z.J.: Spectral finite volume method for conservative laws on unstructured grids: basic formulation. J. Comp. Phys. 178, 210–251 (2002)

Performance Comparison of High Resolution Schemes Müslüm Arıcı and Hüseyin Sinasi ¸ Onur

Abstract In the last decades, a great number of high resolution schemes (HRSs) have been proposed to obtain highly accurate and non-oscillatory solutions in the numerical simulation of the flows. In this work, the performance of various HRSs for time-dependent problems is studied. To do this, HRSs are applied to a test problem. In the test problem, advection of rectangular, semi-ellipse, sine-squared and triangular profiles in a constant velocity field is solved. Numerical solutions are obtained for each profile for CFL numbers of 0.98, 0.50, 0.25, 0.10 and 0.05 and for non-dimensional times of 50, 100 and 250. Computations show that some HRSs are not bounded. Also, It is observed that compressive schemes have less error than diffusive schemes and give more satisfactory results particularly for small CFL numbers for time-dependent problems.

1 Introduction Successful modeling of convection-diffusion problems, without introducing excessive artificial dissipation while retaining high-order accuracy, stability and boundedness is a key issue in the numerical simulation of fluid flow problems. The classical first order upwind (FOU) scheme is unconditionally stable but can generate significant numerical diffusion. On the other hand, the higher-order schemes such as second order upwind (SOU) scheme, central differencing scheme (CDS) and QUICK [7] scheme generate more accurate results than FOU but they may lead to nonpyhsical oscillations in regions where a sharp gradient exists. In the last decades, a great number of schemes have been proposed to obtain highly accurate and nonoscillatory solutions in the numerical simulation of the flows. These schemes are called High Resolution Schemes (HRSs) which allow good resolution of steep gradients without introducing oscillations in the solution. HRSs are mainly based on

M. Arıcı (B) Engineering Faculty, Department of Mechanical Engineering, Kocaeli University, 41380 Kocaeli, Turkey e-mail: [email protected]


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M. Arıcı and H.S. ¸ Onur

either Total Variation Diminishing (TVD) which is usually used in compressible flow simulations or Convection Boundedness Criteria (CBC) which is usually used for incompressible flow simulations. In this work, we study the performance of various HRSs for time-dependent advection equation to understand the behavior of them. The HRSs compared in this study are CLAM [15], COPLA [3], CUBISTA [1], EULER [9], HOAB [19], H-QUICK [18], KOREN [18], MINMOD [13, 21], MUSCL [16], OSHER [2], OSPRE [18], SECBC [20], SHARP [10], SMART [6], SMARTER [4], STOIC [5], SUPERBEE [13], SUPER-C [11], UMIST [12], VONOS [17] and WACEB [14].

2 Analysis of the HRSs Following [8], HRSs can be represented in the normalized variable diagram (NVD). The characteristic lines of the HRSs divide the NVD plane into an inner and an outer region, as shown in Fig. 1. While the characteristic lines of the all HRSs except SHARP and SECBC follow the line which passes through the origin (0,0) and (1,1) in the outer region, they differ only in the inner region. As seen in Fig. 1, the characteristic lines of the all HRSs in the inner region of NVD fall in the shaded area which is limited by an upper and a lower line. The schemes whose characteristic lines are close to the upper line (OABCD) and the lower line (OD) are called compressive and diffusive schemes, respectively. In order to compare the performance of the HRSs, an unsteady, one-dimensional, pure advection of rectangular, semi-ellipse, sine-squared and triangular profiles in a constant velocity field is solved. Each of these profiles serves for a certain purpose. The rectangular profile is a basic test to assess if the scheme enables to resolve sharp gradients without numerical diffusion. The semi-ellipse profile enables assessment of the steepening/clipping characteristics of the scheme. The sine-squared which has a local maximum is a relatively smooth profile. The triangular profile is a stringent test problem since it has a sharp maximum and constant gradient in the both steps.

Fig. 1 Inner and outer regions in the NVD

Performance Comparison of High Resolution Schemes

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Table 1 Test profiles Profile Rectangular

φ = 1 for 0 ≤ x ≤ 20Δx φ = 0 elsewhere

Semi-elipse

φ=

1−

(x−xc )2 (10Δx)2

for |x − xc | ≤ 10Δx

φ = 0 elsewhere Sin-squared

φ =sin2

πx 20Δx

for

0 ≤ x ≤ 20Δx

φ = 0 elsewhere Triangular

φ= φ=

x 10Δx 20Δx−x 10Δx

for

0 ≤ x ≤ 10Δx

for

10 ≤ x ≤ 20Δx

φ = 0 elsewhere

The initial profiles are given in Table 1. The width of all profiles is taken as 20 Δx. Unsteady, one-dimensional, pure advection of a scalar φ at constant velocity u is described by ∂φ ∂φ +u =0 ∂t ∂x

(1)

The solution of this equation is φ(x, t) = φ(x − ut, 0) which means that the scalar profile should preserve its initial shape at any time since there is no diffusion. Time and distance are non-dimensionalized, respectively, as follows: t x/u x − xo − ut L= Δx T =

(2) (3)

where x is independent coordinate, xo is initial position, t is time and Δx is mesh width. In order to compare the accuracy of the HRSs quantitatively, the following deviation is defined: ε=

|φexact − φcomputed |

(4)

First-order Euler forward differencing is used for temporal discretization. Numerical solutions for each profile are obtained for CFL numbers of 0.98, 0.50, 0.25, 0.10 and 0.05 and for non-dimensional times of 50, 100 and 250.

530


3 Results and Discussion Computations show that COPLA, EULER, HOAB, H-QUICK, OSPRE, SECBC, SHARP, SMART, SMARTER, STOIC and VONOS schemes are not monotonic in at least one of the profiles. For instance, OSPRE and SMARTER schemes cause undershoots and/or overshoots in rectangular and semi-ellipse profiles, COPLA scheme suffers from unbounded results in the rectangular, semi-ellipse and sinesquared profiles, and EULER, HOAB, H-QUICK, SECBC, SHARP, SMART, STOIC and VONOS schemes cause the same problem in the all profiles. Therefore, these schemes are excluded from the further investigation. CLAM, CUBISTA, KOREN, MINMOD, MUSCL, OSHER, SUPERBEE, SUPER-C, UMIST and WACEB schemes generate bounded results for all cases. The deviation of these schemes for each profile is given in Table 2 for different CFL numbers and nondimensional times. When the deviation of the schemes is inspected for the all cases, it is seen that the position of the schemes in the rank varies depending on the profile, CFL number and non-dimensional time. For example, while SUPER-C is in the first position in the rank for all the cases of conveying rectangular and triangular profiles, it moves down in the rank in semi-ellipse and sine-squared profiles for large CFL numbers and small non-dimensional time. Nevertheless, the bounded schemes can be categorized into four groups. While SUPER-C and SUPERBEE which are in the top of rank for almost all the cases, MUSCL, KOREN and WACEB whose rank varies between third and fifth are in the second group, CUBISTA, CLAM and UMIST are in the third group and OSHER and MINMOD are in the last group. As seen from Table 2, there are very small differences between the errors of schemes for CFL=0.98, since the characteristic lines of the all schemes are close to the FOU line. For instance, the difference between the errors of UMIST and MINMOD schemes which are the first and the last schemes in the rank,respectively, is smaller than 10% for semi-ellipse profile for T = 50. As the CFL number decreases the error of schemes increases. However, increase in the error differs from scheme to scheme. For example, when the error of the schemes for T = 50 and T = 250 is compared for the rectangular profile, it is seen that increase in the error of MINMOD, MUSCL, SUPERBEE and SUPER-C is 76, 52, 7 and 2%, respectively. Therefore, the difference between the compressive and diffusive schemes becomes more evident for the small CFL numbers and particularly large non-dimensional times. This is because compressive schemes form a certain shape right after the solution starts and distort the profile very slightly with increasing time. However, the diffusive schemes distort the profile very significantly with increasing time due to high artificial viscosity.


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Table 2 Error comparison of schemes T = 50 T = 250 0.98 0.50 0.25 0.1

0.05

0.98 0.50 0.25 0.1

0.05

Rectangular CLAM CUBISTA KOREN MINMOD MUSCL OSHER SUPERBEE SUPERC UMIST WACEB

CFL

0.984 0.989 0.954 1.076 0.942 0.978 0.852 0.850 1.000 0.961

2.807 2.876 2.580 3.848 2.413 3.314 1.719 1.266 3.068 2.730

3.164 3.106 2.753 4.465 2.708 3.918 1.690 1.303 3.547 2.906

3.376 3.178 2.819 4.767 2.911 4.242 1.735 1.249 3.804 2.938

3.450 3.195 2.842 4.860 2.985 4.346 1.747 1.227 3.889 2.941

1.774 1.814 1.694 2.136 1.612 1.812 1.342 1.337 1.828 1.746

4.322 4.678 4.145 6.788 3.579 5.721 1.753 1.266 5.286 4.542

4.920 4.873 4.236 7.875 4.127 6.823 1.820 1.331 6.121 4.680

5.355 4.828 4.230 8.444 4.605 7.416 1.841 1.267 6.579 4.591

5.513 4.795 4.241 8.624 4.786 7.604 1.847 1.247 6.730 4.541

Semi-ellipse CLAM CUBISTA KOREN MINMOD MUSCL OSHER SUPERBEE SUPERC UMIST WACEB

0.383 0.399 0.389 0.416 0.383 0.412 0.399 0.398 0.381 0.394

1.272 1.410 1.287 1.952 1.134 1.713 1.400 1.357 1.420 1.414

1.454 1.469 1.343 2.365 1.314 1.984 1.453 1.325 1.701 1.456

1.594 1.466 1.359 2.579 1.432 2.180 1.452 1.264 1.855 1.438

1.641 1.468 1.366 2.647 1.483 2.248 1.451 1.242 1.905 1.428

0.736 0.808 0.750 0.928 0.694 0.858 0.769 0.768 0.754 0.790

2.076 2.329 2.032 4.338 1.644 3.193 1.475 1.476 2.990 2.273

2.517 2.477 2.021 5.377 1.958 4.235 1.480 1.428 3.674 2.328

2.861 2.437 1.972 5.978 2.290 4.827 1.483 1.424 4.083 2.248

3.003 2.406 1.966 6.167 2.425 5.016 1.485 1.386 4.231 2.205

Sine-squared CLAM CUBISTA KOREN MINMOD MUSCL OSHER SUPERBEE SUPERC UMIST WACEB

0.082 0.105 0.094 0.094 0.074 0.058 0.101 0.101 0.078 0.104

0.517 0.623 0.476 1.161 0.282 0.925 0.335 0.233 0.616 0.611

0.696 0.710 0.451 1.670 0.428 1.342 0.411 0.259 0.940 0.617

0.838 0.763 0.463 1.963 0.572 1.593 0.430 0.275 1.176 0.591

0.902 0.781 0.468 2.056 0.628 1.675 0.423 0.278 1.253 0.576

0.324 0.375 0.334 0.430 0.273 0.279 0.295 0.295 0.325 0.368

1.818 2.085 1.523 4.189 1.058 3.011 0.821 0.349 2.613 1.896

2.465 2.467 1.663 5.234 1.547 4.137 0.917 0.315 3.465 2.082

2.905 2.571 1.687 5.749 2.000 4.697 0.913 0.341 3.941 2.048

3.063 2.592 1.697 5.901 2.159 4.863 0.891 0.347 4.089 2.018

Triangular

0.196 0.193 0.181 0.216 0.183 0.218 0.182 0.181 0.193 0.182

0.850 0.874 0.691 1.136 0.645 1.007 0.628 0.385 0.875 0.769

0.927 0.976 0.745 1.296 0.715 1.162 0.720 0.381 0.963 0.843

0.966 0.993 0.745 1.427 0.757 1.232 0.751 0.360 1.022 0.850

0.990 0.994 0.738 1.489 0.772 1.279 0.745 0.342 1.051 0.845

0.419 0.429 0.390 0.561 0.371 0.494 0.384 0.383 0.424 0.405

1.357 1.534 1.198 3.178 0.918 2.283 1.163 0.625 1.870 1.406

1.785 1.817 1.251 4.153 1.169 3.220 1.244 0.626 2.549 1.501

2.102 1.906 1.276 4.632 1.442 3.736 1.249 0.602 2.987 1.477

2.241 1.920 1.274 4.774 1.583 3.891 1.237 0.578 3.128 1.468

CLAM CUBISTA KOREN MINMOD MUSCL OSHER SUPERBEE SUPERC UMIST WACEB

532


4 Conclusions The main findings in this study are summarized as follows: • COPLA, EULER, HOAB, H-QUICK, OSPRE, SECBC, SHARP, SMART, SMARTER, STOIC and VONOS schemes are not monotonic in at least one of the profiles for time-dependent flow simulations. • CLAM, CUBISTA, KOREN, MINMOD, MUSCL, OSHER, SUPERBEE, SUPER-C, UMIST and WACEB schemes generate bounded results for all cases. • HRSs whose characteristic lines are close to each other have similar behaviors. • The position of the schemes in the rank varies depending on the profile, CFL number and non-dimensional time. • As the time increases, the increase in the error of compressive schemes is much smaller than that of diffusive schemes. It can be concluded that the HRSs whose characteristic line is close to the upper line in the NVD, so-called compressive schemes, have less error than the HRSs whose characteristic line is close to the lower line in the NVD, so-called diffusive schemes, and give much more satisfactory results particularly for small CFL numbers for time-dependent problems.

References 1. Alves, M.A., Oliveria, P.J., Pinho, F.T.: A convergent and universally bounded interpolation scheme fort the treatment of advection, Int. J. Numer. Meth. Fl. 41, 47–75 (2003) 2. Chakravarthy, S.R., Osher, S.: Numerical experiments with the Osher upwind scheme for the Euler equation, AIAA Journal. 21, 1241–1248 (1983) 3. Choi, S.K., Nam, H.Y., Cho M.: A high resolution and bounded convection scheme, KSME J.. 9(2), 240–255 (1995) 4. Choi, S.K., Nam, H.Y., Cho, M.: A comparison of high-order convection schemes, Comput. Method Appl. M. 121, 281–301 (1995) 5. Darwish, M.S.: A new high-resolution scheme based on the normalized variable formulation, Numer. Heat Tr. B-Fund. 24, 353–371 (1993) 6. Gaskell, P.H., Lau, A.K.C.: Curvature-Compensated convective transport: SMART, a new bounded-preserving transport algorithm, Int. J. Numer. Meth. Fl. 8, 617–641 (1988) 7. Leonard, B.P.: A stable and accurate convective modeling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Eng. 19, 59–98 (1979) 8. Leonard, B.P.: A survey of finite differences with upwinding for numerical modeling of the incompressible convection diffusion equation, Comput. Tech. Trans. Turbulent Flow, Pineridge, Swansea, Wales, 2, 1–35 (1981) 9. Leonard, B.P.: Locally modified QUICK scheme for highly convective 2-D and 3-D flows, Proceedings of the Fifth International Conference on Numerical Methods in Laminar ad Turbulent Flows, 15, Pineridge Press, Swansea, UK, 5, Part 1, 35–47 (1987) 10. Leonard, B.P.: Simple high-accuracy resolution program for convective modeling of discontinuities, Int. J. Numer. Meth. Fl. 8, 1291–1318 (1988) 11. Leonard, B.P.: The ULTIMATE conservative difference scheme applied to unsteady onedimensional advection, Comput. Method Appl. M. 88, 17–74 (1991) 12. Lien, F.S., Leschzier, M.A.: Upstream monotonic interpolation for scalar transport with application to complex turbulent flows. Int. J. Numer. Meth. Fl. 19, 527–548 (1994)


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13. Roe, P.L.: Some contributions to the modeling of discontinuities flows, Lect. Appl. Math. 22, 163–193 (1985) 14. Song, B., Liu, G.R., Lam, K.Y., Amano, R.S.: On a higher-order bounded discretizationscheme, Int. J. Numer. Meth. Fl. 32, 881–897 (2000) 15. Van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14, 361–370 (1974) 16. Van Leer, B.: Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101 (1979) 17. Varonos, A., Bergeles, G.: Development and assessment of a variable-order non-oscillatory scheme for convection term discretization, Int. J. Numer. Meth. Fl. 26, pp. 1–16 (1998) 18. Waterson, N.P.: Development of a bounded higher-order convection scheme for general industrial applications, Project Report 1994–33, von Karman Institute for Fluid Dynamics. SintGenesius-Rode, Belgium. (1994) 19. Wei, J.-J., Yu, B., Tao, W.-Q., Kawaguchi, Y., Wang, H.-S.: A new high-order accurate and bounded scheme for incompressible flow. Heat Tr. B-Fund. 43, 19–41 (2003) 20. Yu, B., Tao, W.-Q., Zhang, D.S., Wang, Q.W.: Discussion on numerical stability and boundedness of convective discretized scheme, Numer. Heat Tr. B-Fund. 40, 343–365 (2001) 21. Zhu, J., Rodi, W.: A low dispersion and bounded convection scheme, Comput. Meth. Appl. Mech. Eng. 92, 87–96 (1991)

Solving the Convective Transport Equation with Several High-Resolution Finite Volume Schemes: Test Computations Alexander I. Khrabry, Evgueni M. Smirnov, and Dmitry K. Zaytsev

Abstract Results of a comparative study of performance of three high-resolution schemes suggested in the literature for solving the convective transport equation are presented. Using an in-house finite-volume unstructured-grid code, extended computations were performed with the HRIC, CICSAM and M-CICSAM schemes for various two-dimensional test problems of species convection in a prescribed velocity field. Effects of the time discretization method, time step and computational grid on conservation of a species spot shape were analyzed. For all the test cases considered, M-CICSAM technique demonstrated its superiority over the other schemes examined.

1 Introduction There are many areas of CFD applications where accurate numerical solution of the convective (or convection-dominated) transport equation is demanded. Examples are advection of species in environmental flows, material processing, analytical devices, etc. In the Volume-of-Fluid (VOF) method widely used for numerical simulation of free surface flows, the convective transport equation describes the evolution of the volume fraction of fluid in grid cells, and the quality of numerical schemes used for approximation of this equation affects directly the free surface artificial smearing or/and deformation. Two high-resolution schemes, HRIC [2] and CICSAM [4], have become popular last decade for solving the convective transport equation in VOF method. Recently a promising modification of CICSAM scheme, M-CICSAM [5], was suggested. The present work presents the results of a comparative study of the performance of the above-mentioned schemes via solving various 2D test problems of a species spot convection in a prescribed velocity field. A particular attention is paid also to the effect of the time-discretization method applied. D.K. Zaytsev (B) St. Petersburg State Polytechnic University, St. Petersburg 195251, Russia e-mail: [email protected]


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2 Mathematical Formulation In the framework of VOF method, the fluid free surface is reconstructed from the distribution of the fluid volume fraction, C, that equals zero in “empty” grid cells and unity in the cells completely filled with the fluid. The fluid fraction is governed by the convection transport equation (1). A proper discretization of this equation should maintain the step interface profile, i.e. prevent the interface smearing and provide for the solution boundedness. ∂C dC ≡ + ν · ∇C = 0 dt ∂t

(1)

When the finite volume technique is used for the spatial discretization of Eq. (1), one needs to compute the cell-face value of the volume fraction, C f , from the neighboring cell values via some interpolation. In 1D framework, taking into account the flow direction, such interpolation normally involves the values from the donor (D) acceptor (A) and upwind (U) cells, see Fig. 1a. Various interpolation schemes can be formulated in terms of the normalized variable, C, as a relation between C f and CD , see (2). Several well-known interpolation schemes are shown graphically in the Normalized Variable Diagram (NVD), Fig. 1b. The shaded region in the plot corresponds to the Convective Boundedness Criterion (CBC) of the solution, (3), formulated in [1] (α is the Courant number). C f = (1 − λ f )C D + λ f C A ,

λf =

C f − CD 1 − CD

,

C≡

C − CU C A − CU

(2)

According to [1], any interpolation scheme satisfying CBC condition (3) provides smooth (bounded) solution of 1D convection problem (1) by means of the explicit time discretization. One can expect that upwind schemes should smear the interface due to numerical diffusion. Contrary to this, downwind schemes should

Fig. 1 Definition of the cell-face value (a) and the Normalized Variable Diagram (b) with various interpolation schemes: the upwind (UD), downwind (DD) and central (CD) differencing, the 2nd order upwind (SOU), and QUICK

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exhibit some anti-diffusive (or compressive) properties and, hence, sharpen the interface as preferred in VOF method.
1 C D C f min 1, C D α , 0 C D 1

(3)

Apparently, the most sharpening scheme satisfying CBC condition corresponds to the upper bound of the CBC region in NVD plot (Fig. 1b). Such scheme was suggested in [1] and named HYPER-C. Numerical tests have shown however that, HYPER-C scheme yields a bounded solution, sharpens the interface and, overall, exhibits a fairly good performance provided that the flow is normal to the interface; otherwise the scheme tends to wrinkle the interface. So this scheme is not really suitable for solving the fluid-fraction equation (1) in a general case. To obtain a high-resolution scheme with better properties one can try to combine some upwind and downwind schemes taking into account the Courant number, the interface orientation, etc. Several schemes of that kind have been reported in the literature. Examined in the present work, schemes CICSAM [4], HRIC [2], and M-CICSAM [5] are formulated as follows. CICSAM scheme (4) is a combination of HYPER-C (HC) and another compressive scheme, Ultimate-Quickest (UQ) that, in turn, is a combination of HYPER-C, QUICK, and UD. Definition of the interface orientation angle, θ f , is illustrated in Fig. 2a; NVD image of the scheme is shown in Fig. 2b. CICSAM

HC

UQ

= γ f C f + (1 − γ f ) C f , γ f = (cosθ f )2 K CD 6C D + 3 HC UQ HC C f = min , 1 , C f = min αC D + (1 − α) , Cf α 8 (4) ∗ HRIC scheme (5) is constructed in three stages. First, the face value of C corresponds to HYPER-C scheme at α = 0.5. Then, to prevent wrinkling of tangential ∗∗ interface by HYPER-C, the upwind scheme (UD) is introduced into C depending on the interface orientation. Finally, the scheme is corrected so that to fit CBC. Cf

Fig. 2 Interface orientation angle (a) and the NVD plot with CICSAM scheme (b)

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H RIC Cf

⎧ ∗∗ ⎪ ⎨C f , = CD ,

⎪ ⎩ C + C ∗∗ − C D

∗∗

D

f

∗

α < 0.3 α > 0.7 0.7−α 0.7−0.3 ,

0.3 α 0.7

∗

C f = γ f C f + (1 − γ f ) C D ,

C f = min(2C D , 1),

(5) 1/2 γ f = cosθ f

M-CICSAM scheme (6) is positioned by the authors as a combination of best features of HRIC and CICSAM. Note that definition of the interface orientation factor γ f in the three schemes is different. M−C I C S AM

Cf

∗

F ROM M

= γ f C f + (1 − γ f ) C f

∗

C f = min(2C D , 1),

F ROM M

Cf

γ f = |cos θ f |1/4

HC = min 0.25 + C D , C f

,

(6)

3 Test Computations To evaluate the quality of the schemes under consideration, several 2D test problems were solved using an in-house finite-volume unstructured-grid code. Namely, advection and deformation of a “spot” of a passive species (the volume-of-fluid variable) in a prescribed velocity field was simulated using different grids and timediscretization techniques. Conservation of the interface shape was the main criterion for the scheme assessment. As an example, Fig. 3 demonstrates a strong effect of the time-discretization method on the final shape of a hollow-square spot carried by a uniform inclined flow. The traveled distance is about 7 times the square edge. The results presented were obtained with CICSAM scheme using a uniform Cartesian grid (40 cells per the square edge); the other schemes displayed quite similar behavior. One can conclude that the Crank–Nicolson scheme is definitely preferable (as compared to the 1st order explicit and implicit schemes) for solving the convective transport equation (1). All the results presented below were obtained with this scheme. For the same flow and grid configuration, Fig. 4 presents the results obtained with the three schemes examined at a higher Courant number (α = 0.75). One can observe a considerable smearing of the spot boundary by CICSAM and (even more) HRIC, whereas M-CICSAM scheme produces a quite satisfactory result.

Fig. 3 Artificial deformation of the hollow-square spot in a uniform flow: the explicit (a), implicit (b), and the Crank–Nicolson (c) time-discretization scheme (CICSAM, α = 0.25)

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Fig. 4 Smearing of the boundary of the hollow-square spot in a uniform flow: HRIC (a), CICSAM (b), and M-CICSAM (c) scheme (Crank–Nicolson, α = 0.75)

Fig. 5 Final shape of the hollow-square spot computed with CICSAM (a) and M-CICSAM (b) scheme using triangular-cell grid (c)

The data presented in Fig. 5 were obtained with the time step and grid spacing identical to those in Fig. 3 but using the triangular-cell grid. One can see a weak wrinkling of the spot boundary, less pronounced for M-CICSAM scheme. One more conclusion suggested by the triangular-cell-grid test is that the spot boundary may tend to align with grid cell faces. To investigate this aspect in more detail, advection of a circular spot in a uniform flow was simulated using a rather coarse Cartesian grid (12 cells per the circle diameter). The traveled distance was about 8 times the diameter. The results obtained at the Courant number of 0.3 are presented in Fig. 6. Evidently, HRIC and CICSAM schemes failed to preserve the spot shape (that became nearly square), whereas M-CICSAM scheme proved its advantage once again. The results obtained with M-CICSAM scheme for a more complicated test problem described in [3] are presented in Fig. 7. Here the velocity components of the carrier flow are defined as u = sin(x) cos(y), v = − cos(x) sin(y). The initial condition of the spot is a circle of radius 0.2π with center at point (0.5π, 0.2(π + 1)). The computational domain (a square of side π) is covered by uniform Cartesian

Fig. 6 Final shape of the initially round spot obtained with M-CICSAM (a), CICSAM (b), and HRIC (c) scheme

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Fig. 7 Spot shape after the forward (a) and backward (b) deformation in a shear flow (α = 0.25, 1000+1000 time steps, M-CICSAM)

grid 100×100 cells. The integration time step equals π/400 (the maximum Courant number about 0.25). The test essence was as follows. First, the spot was transported during a prescribed time period (1000 time steps for the case presented) that resulted in a considerable deformation of the spot (Fig. 7a). After that, the flow was reversed and the spot was transported back for the same time period. Ideally, after the forward and the backward deformation, the spot should restore its initial shape. The data presented in Fig. 7b evidence that the M-CICSAM prediction is nearly perfect. Summarizing the above, one can conclude that for all the test cases considered the M-CICSAM scheme [5] has demonstrated its superiority over the other highresolution schemes examined. Acknowledgements The work was partially supported by the Russian Foundation of Basic Research (grant 08-08-00977).

References 1. Leonard, B.P.: The ULTIMATE conservative difference scheme applied to unsteady one dimensional advection. Comput. Methods Appl. Mech. Eng. 88, 17–74 (1991) 2. Muzaferija, S., Peric, M., Sames, P., Schelin, T.: A two-fluid Navier-Stokes solver to simulate water entry. Proceedings of the 22 Symposium on Naval Hydrodynamics, pp. 277–289, Washington, DC (1998) 3. Rudman, M.: Volume-tracking methods for interfacial flow calculations. Int. J. Numer. Meth. Fluids 24, 671–691 (1997) 4. Ubbink, O., Issa, I.: A method for capturing sharp fluid interfaces on arbitrary meshes. J. Comput. Phys. 153, 26–50 (1999) 5. Waclawczyk, T., Koronowicz, T.: Remarks on prediction of wave drag using VOF method with interface capturing approach. Arch. Civ. Mech. Eng. 8(1), 5–14 (2008)

Part XX

Free-Surface Flow

Computational Study of Hydrodynamics and Heat Transfer Associated with a Liquid Drop Impacting a Hot Surface Edin Berberović, Ilia V. Roisman, Suad Jakirlić, and Cameron Tropea

Abstract The present work deals with computational modeling of the fluid flow and heat transfer taking place in the process of impact of a cold liquid drop onto a dry heated substrate. The computational model, based on the volume-of-fluid method for the free-surface capturing, is validated by simulating the configurations accounting for the conjugate heat transfer. The simulations were performed at different impact Reynolds and Weber numbers. The considered temperature range of the drop-surface, i.e. liquid-solid system doesn’t account for the phase change, that is boiling and evaporation. The model performances are assessed by contrasting the results to the reference database originating from the available experimental investigations. Contrary to some previous numerical studies, the present computational model accounts for the air flow surrounding the liquid drop. The reported results agree reasonably well with experimental and theoretical findings with respect to the drop spreading pattern and associated heat flux and temperature distribution.

1 Introduction Drop impingement onto a hot surface represents a complex phenomenon influenced by drop size and velocity, surface temperature and roughness, angle of impact, liquid properties, liquid wall film generated in spray impact or eventual phase change. Understanding the non-isothermal drop impact still remains incomplete due to its high unsteadiness and small scales involved, inhibiting direct experimental access where high-speed photography is still a primary source of information. The main goal of the present study is to formulate a computational model for conjugate heat transfer in the framework of the VOF-based volume tracking [3]. Single water drop impact onto heated stainless steel wall without phase change investigated experimentally in [8] is used for validation. The governing dimensionless parameters are S. Jakirlić (B) Institute of Fluid Mechanics and Aerodynamics/Center of Smart Interfaces, Technische Universität Darmstadt, D-64287 Darmstadt, Germany e-mail: [email protected]


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the impact Reynolds, Weber and Prandtl numbers, Re = D0 U0 /ν, We = ρ D0 U02 /σ and Pr = k/(ρc p ), where ρ, σ , ν, k and c p are the liquid density, surface tension, kinematic viscosity, thermal conductivity and specific heat, respectively and D0 and U0 are the impact drop diameter and velocity. The dimensionless time is t = tU0 /D0 .

2 Computational Model Governing equations for the fluid flow and heat transfer due to the drop impact process include transport equations for conservation of mass, phase fraction, momentum and energy for fluid and solid regions ∇ · U = 0,

(1)

∂γ + ∇ · (Uγ ) + ∇ · [Ur γ (1 − γ )] = 0, ∂t

(2)

∂(ρU) + ∇ · (ρUU) = −∇ pd − g · x∇ρ + ∇ · T + σ κ∇γ , ∂t ∂(ρc p T ) + ∇ · (ρc p UT ) = ∇ · [k∇(T )], ∂t ∂(ρs c p,s Ts ) = ∇ · [ks ∇(Ts )], ∂t

(3) (4) (5)

where the symbols are velocity U, liquid phase fraction γ , liquid-gas relative velocity Ur , modified pressure pd = p − ρg · x, gravity acceleration g, position vector x, stress tensor T = μ(∇U + (∇U)T ), surface curvature κ, fluid temperature T , respectively, and subscript s refers to the solid substrate. The fluid is a homogeneous mixture of Newtonian immiscible fluids, with properties evaluated as weighted averages. Surface tension effects are included as body force according to the Continuum Surface Force model (CSF) [2] neglecting forces resulting from spatial variation of surface tension. The interface shape is determined from the solution of the phase fraction equation, taking values between zero (gas) and one (liquid). Viscous dissipation is neglected in the fluid energy equation, air and steel properties are assumed constant, whereas water properties depend on temperature and are evaluated using expressions obtained by regression performed on the available property data ν = −2, 61 · 10−12 T 3 + 5.82 · 10−10 T 2 − 4.68 · 10−8 T 1 + 1.74 · 10−6 , k = −9, 74 · 10

−6

T + 2, 12 · 10 2

−3

−1

T + 5, 58 · 10

,

(6) (7)

The computations were carried out using the open source code OpenFOAM of OpenCFD Ltd. The phase fraction γ changes smoothly across the interface and the smearing of steep gradients is suppressed by the additional convective term in the phase fraction equation (2). The relative flux appearing in the term is replaced with the compression flux acting normal to the interface [11] and active only within the

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thin interface region, the details of implementation given in [1]. The computational domain is fixed 2D axisymetric ∼ 5D0 × 5D0 in vertical plane, with a graded mesh having the best resolution in the region of liquid motion. The mesh has 70,000 cells in total, 4/7 belonging to the fluid and the rest to the solid region. Fluid flow in the region above the solid surface is iteratively solved followed by solving the energy equation in a coupled manner for both regions simultaneously. To this end the discretization matrix for the energy equation is appropriately extended including the coefficients’ contributions from the entire solid domain as well, enabling heat fluxes across the fluid-solid interface to be internally conserved. Finite volume approximations on the collocated mesh and self-adjusting time step are used. Transient terms are evaluated using Euler implicit scheme, spatial derivative terms are integrated over cell surfaces, convection is approximated using flux limiters [5, 10] and the velocity-pressure coupling is based on the PISO algorithm [4]. At the fluid-solid interface no-slip wall boundary is set and other boundaries are open. For the temperature, fixed values are set at both regions with adiabatic side boundary.

3 Results and Discussion Initial water and steel properties and the governing dimensionless numbers listed in Table 1 correspond to two cases in [8]. The initial drop-solid temperature difference is (Ts − Td )0 = (120 − 25)◦ C. Prandtl number evaluated at the analytically determined contact temperature Tc = 98.6◦ C is Pr = 1.81, [9]. Present numerical results for spread factor are compared with experimental ones in Fig. 1. The agreement is very good during spreading and the overprediction in the receding phase is attributed to receding contact angles. In [8] the receding contact angles and the functional forms of the temperature dependent properties are not given. For the present purposes, the contact angle has the value of 110◦ for the advancing [8] and 40◦ for the receding phase [7]. In accordance with previous findings a smaller spread factor in case of constant properties is predicted. In VOF-related methods, air bubbles entrapped at the impact point are commonly encountered. The model in [8] solves liquid flow only, neglecting the presence of air, thus being incapable of reproducing entrapped air bubbles. In the present and some other studies [6] such bubbles were captured. Computed spreading drop pattern and temperature distribution within the fluid and the substrate at several times are displayed in Fig. 2, normalized using the initial drop-solid temperature difference. The distributions are similar to those computed in [8], except for the impact region where Table 1 Thermophysical properties and characteristic dimensionless numbers ν, m2 s−1 Steel Water, A 0.894·10−6 Water, B 1.004·10−6

σ , N/m ρ, kg m−3 0.072 0.0728

7, 900 997.1 998.2

k, W/(mK) c p , J/(kg K) Re 16.6 0.59 0.6

515 4,182 4,179

We

Pe

2,908 47 5,263 4,474 111 8,098

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D

3

2 variable properties constant properties case A case B

1

0

0

1

2

3

4

5

t

Fig. 1 Time evolution of the spreading drop diameter: comparison of the present numerical predictions with experimental results from [8]

t = 0.5 ms

t = 1 ms

t = 2 ms

t = 3 ms T 0.85 0.9 0.95

T 0.2 0.4 0.6 0.8

Fig. 2 Spreading drop pattern and corresponding temperature distributions at different time instants (case A)

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130 experimental numerical: variable properties constant properties re-initialized

Tw , °C

120 110 100 90 80

0

1

2 t, ms

3

4

5

Fig. 3 Comparison of the present numerical predictions with experimental results for the temporal evolution of the impact point temperature from [8] (case A)

130

10.0

120

8.0 qw , MW/m2

110 100

·

Tw , °C

the entrapped bubble is presently observed, yielding higher temperatures. In Fig. 3 numerical predictions are contrasted to experimental results for impact point temperature. The overprediction of the experiment by the simulation is clearly due to the presence of a bubble at the impact region. An additional hypothetical case with same impact parameters was computed by first stopping the simulation at the moment of impact, re-initializing the distribution of the phase fraction in cells occupied by the bubble and then continuing the simulation. The impact point temperature agrees very well with experiments in the latter case, however this is only a hypothetical case used to examine the performance of the numerical code and is not considered as relevant for the overall analysis of heat transfer. Computed radial distributions of temperature and heat flux at the solid surface are presented in Fig. 4. The agreement with results of [8] is fairly good except in the vicinity of the impact region, where higher temperatures and smaller heat fluxes are encountered, with small peaks due to the entrapped air bubble.

90 80

6.0 4.0 2.0

t = 0.5 ms t = 1 ms t = 3 ms

0.0 0

2

1 r, mm

t = 0.5 ms t = 1 ms t = 3 ms

3

0

1

2

3

r, mm

Fig. 4 Radial distributions of temperature (left) and heat flux (right) at the solid surface: comparison of the present predictions with numerical results from [8](case A)

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The distributions are rather smooth in the reminder of the observed area up to the rim region, where domains of higher and lower temperatures are created due to the local flow field.

4 Conclusions The potential of the computational model for conjugate heat transfer in the VOFbased framework was illustrated by computing single water drop impact onto a steel surface. The change in the drop morphology quantified in terms of spreading diameter was returned in good agreement with reference experiment. The artificial removal of the entrapped air bubble in the computational procedure led to the experimentally determined temperature values. The afore-mentioned air bubble entrapment is the consequence of the air flow considered simultaneously. The present computational model enables a more realistic insight into the heat transfer process relevant also to the spray cooling, where situations with air entrainment within the wall film may easily arise.

References 1. Berberović, E., vanHinsberg, N., Jakirlić, S., et al.: Drop impact onto a liquid layer of finite thickness: Dynamics of the cavity evolution. Phys. Rev. E. 79(3)(036306), 1–15 (2009) 2. Brackbill, J., Kothe, D., Zemach, C.: A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354 (1992) 3. Hirt C., Nichols B.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225 (1981) 4. Issa, R.: Solution of the implicitly discretised fluid flow equations by operator-splitting. J. Comput. Phys. 62, 40–65 (1986) 5. Jasak, H., Weller, H., Gosman, A.: High resolution nvd differencing scheme for arbitrarily unstructed meshes. Int. J. Numer. Methods Fluids 31, 431–449 (1999) 6. Mehdi-Nejad, V., Mostaghimi, J., Chandra, S.: Air bubble entrapment under an impacting droplet. Phys. Fluids 15, 173–183 (2003) 7. Pasandideh-Fard, M., Aziz, S., Chandra, S., et al.: Cooling effectiveness of a water drop impinging on a hot surface. Int. J. Heat Fluid Flow 22, 201–210 (2001) 8. Pasandideh-Fard, M., Qiao, Y., Chandra, S., et al.: Capillary effects during droplet impact on a solid surface. Phys. Fluids 8, 650–659 (1996) 9. Roisman, I.: Fast forced liquid film spreading on a substrate: Flow, heat transfer and phase transition. J. Fluid Mech. 656, 189–204 (2010) 10. Van Leer, B.: Towards the ultimate conservative difference scheme II. Monotonicity and conservation combined in a second order scheme. J. Comput. Phys. 14, 361–370 (1974) 11. Weller, H.: A new approach to VOF-based interface capturing methods for incompressible and compressible flow. Tech. Rep. TR/HGW/04, OpenCFD Ltd (2008)

A Second Order JFNK-Based IMEX Method for Single and Multi-Phase Flows Samet Kadioglu, Dana Knoll, Mark Sussman, and Richard Martineau

Abstract We present a second order time accurate IMplicit/EXplicit (IMEX) method for solving single and multi-phase flow problems. The algorithm consists of a combination of an explicit and an implicit blocks. The explicit block solves the non-stiff parts of the governing system whereas the implicit block operates on the stiff terms. In our self-consistent IMEX implementation, the explicit part is always executed inside the implicit block as part of the nonlinear functions evaluation making use of the Jacobian-free Newton Krylov (JFNK) method (Knoll and Keyes, J. Comput. Phys. 193:357–397, 2004). This leads to an implicitly balanced algorithm in that all non-linearities due to the coupling of different time terms are consistently converged. In this paper, we present computational results when this IMEX strategy is applied to single/multi-phase incompressible flow models.

1 Introduction The IMplicit/EXplicit (IMEX) time integration technique has been increasingly used in physical applications that consist of multiple physics or exhibit multiple time scales [1, 2, 4–6, 9]. In general, the governing equations for these kinds of applications consist of stiff and non-stiff terms. A typical IMEX method separates the stiff and non-stiff parts of the governing system and employs an explicit discretization (explicit block) that solves the non-stiff part and an implicit discretization (implicit block) that solves the stiff part of the problem. In a classic IMEX implementation [1, 2, 9], the time operators are splitted in such a way that the explicit and implicit algorithm blocks are executed independent of each other resulting in non-converged non-linearities therefore time inaccuracies (order reduction to the first order in time accuracy for certain models is often reported [2, 4]). On the S. Kadioglu (B) Idaho National Laboratory, Fuels Modeling and Simulation Department, Idaho Falls, ID 83415, USA e-mail: [email protected]


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other hand, in our self-consistent IMEX implementation, the explicit block is always solved inside the implicit block as part of the nonlinear functions evaluation making use of the Jacobian-Free Newton Krylov (JFNK) method [7]. In this way, there is a continuous interaction between the two algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all the nonlinearities due to the coupling of different time terms are consistently converged. In other words, we obtain an IMEX method that can maintain the formal order of time accuracy of the numerical scheme. We have applied this JFNK-based IMEX methodology to a multi-physics problem that couples a neutron diffusion model to a thermally driven mechanics model to simulate transient behavior of fast burst reactors [6]. We have also considered multiple time scale flow models that consist of the compressible Euler equations plus very stiff heat conduction and/or source terms to solve radiation hydrodynamics problems [4, 5]. For all of these applications, we have successfully showed demonstrated-second-order time convergence. In this paper, we consider incompressible single and multi phase flow models. In particular, the single-phase model consists of the incompressible Navier-Stokes equations and the multi-phase model considers the incompressible Navier-Stokes equations written in terms of the level set function plus the level set interface advection equation [11]. The JFNK-based IMEX method is applied to the single-phase flow model in the following manner. The hyperbolic terms of the flow equations (momentum advection terms except the pressure gradients) are solved explicitly exploiting the well understood explicit schemes. On the other hand, an implicit strategy is employed for the non-hyperbolic terms (viscous and pressure terms). In the multi-phase case, the IMEX method solves the momentum advection plus the interface advection terms explicitly and the viscous plus the stiff surface tension force terms implicitly. The implicit/explicit time integrations are carried out in a similar self-consistent fashion (refer to the first paragraph).

2 Governing Equations The single-phase flow model uses the following non-dimensional incompressible Navier-Stokes equations ut + u · ∇u = −∇ p + ∇ · u = 0,

1 Δu, Re

(1) (2)

where u = (u, v) is the flow velocity, p is the fluid pressure, and Re is the non-dimensional Reynolds number. When we solve multi-phase (gas-liquid) flow systems, we rewrite Eq. (1) in terms of the level set function (φ) and add surface and body forces, i.e.,

A Second Order JFNK-Based IMEX Method

ut + u · ∇u = −

∇p 1 + ∇ · [μ(φ)(∇u + (∇u)T )] + Fs + Fg . ρ(φ) Reρ(φ)

551

(3)

The interface motion is governed by the level set advection equation, φt + u · ∇φ = 0,

(4)

where φ(x, t) > 0 if x ∈ liquid, φ(x, t) < 0 if x ∈ gas, and φ(x, t) = 0 if x ∈ Γ where Γ represents the gas-liquid interface, and is defined as the zero level set of φ. The density and viscosity can be written in terms of the level set function as ρ(φ) = ρg [1 − H (φ)] + ρl H (φ) and μ(φ) = μg [1 − H (φ)] + μl H (φ) where ρg , ρl , μg , and μl represent the gas density, liquid density, gas viscosity, and liquid viscosity respectively. H (φ) is the smoothed Heaviside function [11]. The term Fs = −[1/(W eρ(φ))]κ(φ)∇ H (φ) represents the forcing term due to surface tension with κ(φ) = ∇ · (∇φ/|∇φ|) being the local mean curvature. More information about the derivation and the problem parameters for these equations can be found in [10].

3 The JFNK-Based IMEX Method Assume that we are solving the multi-phase model, then the Explicit Block of IMEX solves advective terms of Eq. (3) (e.g., ut + u · ∇u = 0) plus Eq. (4) whereas the Implicit Block operates on the right hand side of Eq. (3) (e.g., ut = r hs) plus the divergence free constraint (Eq. (2)). Our explicit and implicit time discretizations are based on a second order Runge-Kutta and Crank-Nicolson methods [8]. Our spatial discretizations for the explicit step are based on a second order essentially non-oscillatory (ENO) scheme, and centered differencing for the implicit step [8]. We write our implicit discretization as a system of non-linear equations (F(U ) = 0 where U = (u, v, p)) and employ the JFNK method to solve it. More details about how to form F(U ) and the IMEX algorithm implementations can be found in [4]. We remark again that the explicit step is always solved inside the implicit loop as part of the JFNK’s non-linear functions evaluation. This helps to converge the tightly coupled non-linear system. To increase the efficiency of the JFNK solver, we use the classic pressure projection methodology as the preconditioner. Then the resulting elliptic system is relaxed by a few algebraic multi-grid V-cycles.

4 Numerical Results First, we present results from the single-phase flow calculations. We consider a test problem that consists of a pair of horizontal shear layers of finite thickness, perturbed by a small amplitude vertical velocity. Details about the problem settings can be found in [3]. Figure 1 shows the time history of the numerical solutions calculated in a doubly-periodic unit square with a 256 × 256 mesh. In time, each shear layer evolves into a periodic array of large vortices with the shear layer between the rolls

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Time = 0.8

Time = 1.2

Fig. 1 Vorticity solutions from the thin shear layer problem on 256 × 256 mesh

being thinned by the large straining field there. Eventually, these thinned layers wrap around the large rolls. The evolution of the top and bottom layers are mirror images of one another. Notice that the flow is highly transient throughout the simulation time, so it is a good test to check the numerical time accuracy of the scheme. We provide a time convergence analysis for the X/Y -velocities and pressure in Fig. 2. Figure 2 shows the decrease in the L 2 -norm of the time errors of the numerical solutions. It is clear from this figure that we have obtained second order time convergence for all flow variables. Figure 2 also provides some insights about the quality of our spatial discretization. It is evident form Fig. 2 that the method resolves the complicated flow structure reasonably well. Now, we present results from the multi-phase flow calculations. The problem also studied in [10] consists of a gas (air) bubble rising in an incompressible liquid (water). The bubble center is initially located at (x, y) = (1, 1) in a 2×4 rectangular domain. Figure 3 shows the time history of the bubble motion. The calculations are carried out with 100 × 200 mesh points. Notice that the bubble loses its circular shape in time due to surface forces effects. Also, we note that using a smoothed Heaviside function smears the interface into few cells (diffused interface capturing). Nonetheless, the density ratio can change from 1 to 1,000 for this problem, therefore it is a challenging test for any multi-phase flow solver. Figure 3 indicates that Y−VELOCITY −2 −3

−4 −5 −6 −7 −4

−3

Num. Sol Sec. Order

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log10L2(Error)

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X−VELOCITY Num. Sol Sec. Order log10L2(Error)

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Fig. 2 L 2 norm of the time errors for the thin shear layer problem on 256×256 mesh at time = 1.0

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Fig. 3 Density at time = (starting from left) 0.00, 0.35, 0.43, and 0.51 Table 1 JFNK-based IMEX method versus CLSVOF method JFNK-based IMEX method CLSVOF method Total # time steps

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our method captures the inter-facial structure reasonably well. Moreover, our JFNK solver takes on average 3 Newton and 10 Krylov iterations. Table 1 compares the performance of our method to the CLSVOF method [11] for the same problem. Table 1 indicates that our method takes 11 times less number of time steps to reach the final time = 0.51.

5 Conclusion We have presented a second order time accurate JFNK-based IMEX method for solving single and multi-phase flow problems. In our implicitly balanced IMEX method, all non-linearities due to the coupling of different time terms are consistently converged leading to second order time accurate calculations. The key feature of our method is that we carry out the explicit integrations as part of the non-linear functions evaluation within the JFNK framework. We have provided few examples to verify the accuracy and performance of our method. Acknowledgements The submitted manuscript has been authored by a contractor of the US Government under Contract No. DEAC07-05ID14517 (INL/CON-10-19312).

References 1. Ascher, U.M., Ruuth, S.J. Wetton, B.T.R.: Implicit-explicit methods for time-dependent pde’s. SIAM J. Num. Anal. 32, 797–823 (1995)

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2. Bates, J.W., Knoll, D.A., Rider, W.J., Lowrie, R.B., Mousseau, V.A.:On consistent timeintegration methods for radiation hydrodynamics in the equilibrium diffusion limit: Low energy-density regime. J. Comput. Phys. 167, 99–130 (2001) 3. Bell, J.B., Colella, P., Glaz, H.M.: A second order projection method for the incompressible navier-stokes equations. J. Comput. Phys. 85(2), 257–283 (1989) 4. Kadioglu, S.Y. Knoll, D.A.: A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems. J. Comput. Phys. 229(9), 3237–3249 (2010) 5. Kadioglu, S.Y., Knoll, D.A., Lowrie, R.B., Rauenzahn, R.M.: A second order selfconsistent imex method for radiation hydrodynamics. J. Comput. Phys. 229(22), 8313–8332 (2010) 6. Kadioglu, S.Y., Knoll, D.A., Oliveria, C.: Multi-physics analysis of spherical fast burst reactors. Nucl. Sci. Eng. 163, 1–12 (2009) 7. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397 (2004) 8. Leveque, R.J.: Finite Volume Methods for Hyperbolic Problems. Texts in Applied Mathematics. Cambridge University Press, Cambridge (1998) 9. Lowrie, R.B., Morel, J.E., Hittinger, J.A.: The coupling of radiation and hydrodynamics. Astrophys. J. 521, 432 (1999) 10. Olsson, E., Kreiss, G.: A conservative level set method for two phase flow. J. Comput. Phys. 210, 225–246 (2005) 11. Sussman, M., Puckett, E.G.: A coupled level set and volume-of-fluid method for computing 3d and axisymmetric incompressible two-phase flows. J. Comput. Phys. 162, 301–337 (2000)

Simulation of Two-Phase Flow in Sloshing Tanks

Roel Luppes, Arthur Veldman, and Rik Wemmenhove

Abstract The CFD simulation tool ComFLOW is applied to study the effect of tank motions on two-phase flow phenomena inside a sloshing tank. An improved VOF method is used to assure an accurate description of the fluid displacement. With a novel “gravity-consistent” density averaging method, spurious velocities near the free surface can be avoided. Comparison of simulations with measurements show that compressibility of the air should be included for accurate simulations; the agreement is quite good on a relatively coarse mesh.

1 Introduction The maritime transport of liquid natural gas (LNG) in partially filled tanks grows considerably. This enhances the demand for methods that accurately predict the fluid behaviour inside sloshing tanks. The CFD simulation tool ComFLOW has been developed initially to study sloshing fuel on board spacecraft in micro-gravity, for which a very accurate and robust description of the free surface is required [3, 4, 7, 8]. Later, the methodology was extended to simulations of sloshing liquids and two-phase flow in offshore applications, such as green water loading [2, 7], impact loads on fixed structures [2, 6, 7] and sloshing tanks [5, 10]. ComFLOW solves the Navier-Stokes equations in both water and air, with second order accuracy in both space and time, by means of second-order upwind discretisation in combination with Adams-Bashforth time-stepping. The water surface is advected by means of a modified Volume-of-Fluid (VOF) method, with improved accuracy through a height-function approach. This means that the interface is explicitly reconstructed through a local height function and subsequently advected, ensuring a sharp interface without smearing (see Sect. 3). R. Luppes (B) J. Bernoulli Institute for Mathematics and Computer Science, University of Groningen, 9700 AK Groningen, The Netherlands e-mail: [email protected]


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Fig. 1 Air entrainment during a model test (10% filling rate; irregular sway and roll motion)

Compressibility of the air can be included. This is especially important in cases of violent flow conditions, when interesting two-phase phenomena occur, such as air entrapment and air entrainment (see Fig. 1).

2 Governing Equations The cushioning effect due to the compressibility of air is important in wave impact simulations. Hence, the conservation equations (mass and momentum) for compressible flow should be considered, in terms of time t, velocity u = (u, v, w)T , pressure p and fluid density ρ and viscosity μ [5, 6, 10]. In case of rotating sloshing tanks, besides gravity, the force-term in the equations also contains body forces (e.g. Coriolis force) to account for a rotating reference frame [3, 4, 7, 8]. As a boundary condition, the no-slip condition u = 0 is imposed at solid walls. To include compressibility of air, the Navier-Stokes equations are closed with the equation of state ρg = ρg,0 pg,0 (1−1/γ ) pg 1/γ , where pg,0 and ρg,0 denote the constant atmospheric pressure and ambient gas density, respectively, and γ = 1.4 for pure air. The free surface is described by Fs (x, t) = 0, and its motion is given by a simple advection equation. Boundary conditions at the interface are determined by a force balance (in terms of surface tension σ and mean curvature κ) in case of one-phase flow [2–4, 7, 8] or by an additional body force in case of two-phase flow [5, 6, 10].

3 Numerical Techniques Discretisation The equations are coupled by means of an explicit time-stepping procedure. When time discretisation (time levels n, time step Δt) is applied to the Navier-Stokes equations, several terms arise that need further attention, such as ρ n+1 , p n+1 and the convection term ρ1n ∇ · (ρ n u n u n ). The treatment of unknowns at the new time level n+1 is described below.


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Spatial discretisation is done on a staggered Cartesian grid. The convective term, dominant in momentum-driven applications, is treated with symmetry preserving spatial discretisation [9]. With the 1st-order upwind (B2) scheme, artificial diffusion is applied to obtain stable solutions, leading to abundant damping of fluid motion. Using the 2nd-order upwind (B3) scheme, the level of artificial diffusion is much smaller, resulting in less damping. B2 can be combined with 1st-order Forward-Euler (FE) time discretisation. For B3, the 2nd-order Adams-Bashforth (AB) method is used to obtain stable solutions [3–8, 10]. The combination of B3/AB makes ComFLOW second order accurate in both space and time. Stability is controlled by η = uΔt/Δx and d = 2μΔt/(Δx)2 . In practical offshore cases, convection dominates, and η determines the stability limit. For the combinathe CFL limit is ηmax ≤ 1−d ≈ 1. For B3/AB the limit is more restrictive tion B2/FE, ηmax ≤ 14 − 12 d ≈ 14 ; the price to reduce artificial damping of fluid motion. Poisson equation The pressure p n+1 is calculated from a Poisson equation, derived by combining the discretised Navier-Stokes equations. Initially, this equation contains terms with ρ n+1 and ∇ρ n , besides convective, diffusive and force effects. The treatment of these terms is described in [10]. First, ρ is substituted analytically by the gas density ρg through ρ = Fs ρl + (1 − Fs )ρg , with Fs the liquid fraction in a cell. As a result, derivatives of ρ no longer contain large jumps, as they are only determined by compression or expansion of the gas phase. Next, ρgn+1 is substituted using the polytropic equation of state (see Sect. 2). Finally, p is linearized by a Newton approximation to eliminate the exponent 1/γ and then transferred to the left-hand side of the Poisson equation [5, 10]. The entries in the Poisson matrix ∇ · ρ1n ∇ pn+1 may differ up to a factor 1,000, because of jumps in ρ n (water vs. air). This enhances the need for a powerful solver. The Poisson equation is solved with a Krylov subspace method, using incomplete LU preconditioning for acceleration. Free-Surface displacement In the VOF method for free-surface displacement, filling rates Fs of individual cells are administrated and fluid fragments are advected with local velocities. The free surface position is reconstructed from combined fluid volumes contained in single cells. With Simple Linear Interface Calculation (SLIC), the interface consists of line segments constructed parallel or perpendicular to the major flow axes. A characteristic drawback of SLIC is the unphysical creation of disconnected droplets, resulting from errors in the reconstruction. Moreover, values are rounded off (0 ≤ Fs ≤ 1), leading to significant losses in liquid mass. To avoid isolated droplets and mass losses, a local height function (LHF) has been introduced in ComFLOW. The LHF is applied in a 3 × 3(×3) block (in 2D or 3D) of cells surrounding a Surface-cell. First, the orientation of the free surface is determined (horizontal or vertical), depending on Fs values in the surrounding block of cells. Next, the horizontal or vertical height in each row or column is computed by summing the Fs values (see Fig. 2). Based on the LHF, fluid is transported from one cell (donor) to another (acceptor), depending on the magnitude of velocity, time step and grid sizes. The interface at the new time instant is then reconstructed by means of another LHF. Compared to the original VOF method [1], the LHF approach results in significantly less droplets and water loss [2, 5, 7, 8, 10].

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The difference between simple and sophisticated density averaging for a hydrostatic case is shown in Fig. 3. The effect of the density averaging method on the free surface for a regular sway sloshing experiment with 10% filling ratio is shown in Fig. 4. Naive geometrical averaging clearly results in instabilities, whereas gravityconsistent averaging results in much less disturbances near the free surface.

4 Validation Results To examine the relevant flow phenomena and liquid motion inside LNG carriers, model experiments have been carried out on a 1:10 scale (Fig. 1: 3.9 m × 2.7 m). Various tank filling-ratios and different types of motion have been tested to study the sloshing behaviour at various sea states. The model experiments thus provide extensive validation material for ComFLOW. The behaviour of the sloshing liquid strongly depends upon the regularity of the tank motion and the filling ratio. Video frames, wave probes and pressure transducers have been used to compare the fluid flow of simulations and experiments. Two-phase effects, such as air entrapment, are more common in case of increasing tank filling ratios and irregular tank motions. In Fig. 5, profiles are shown of the water-heights at the tank-center and pressure signals at the lower right corner. Experiments are compared with 3 simulations: a 1-phase simulation (only water; air not included) and two 2-phase simulations, where both water and compressible air are taken into account. The comparisons show that for accurate two-phase simulations, 2nd-order upwind (B3) has to be applied to reduce numerical dissipation (especially of the air), as indicated by the hampered fluid motion in case of two-phase B2. Moreover, for a realistic simulation of two-phase effects, compressibility of the air should be included. In the

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Fig. 5 Water-heights and pressure signals versus time (s), for a case with 10% filling rate and regular sway motion (upper figures) and a case with 25% filling rate and irregular sway and roll motion (lower figures). Experiments (Exp) are compared with 1-phase B2 (1), 2-phase B2 (2a) and 2 phase B3 (2b) simulations on a 195 × 135 grid

one-phase simulation of irregular tank motion (Fig. 5, lower figures), the fluid is not affected by the compressible air, resulting in an overestimation of fluid motion. The agreement between experiments and the most accurate simulations (two-phase, compressible air, B3-scheme) is quite good, despite the use of a relatively coarse 195 × 135 mesh.

5 Conclusions The CFD simulation tool ComFLOW has been applied to study the effect of tank motions on two-phase flow phenomena inside a sloshing tank. The fluid displacement is described using a modified VOF method, with improved accuracy through a height-function approach. With a novel “gravity-consistent” density averaging method, spurious velocities near the free surface can be avoided. Compressibility of the air should be included for realistic simulations of two-phase effects. The agreement between experiments and the most accurate simulations (two-phase, compressible air, B3-scheme) is quite good on a relatively coarse mesh.


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References 1. Hirt, C.R., Nichols, B.D.: Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comp. Phys. 39, 201–225 (1981) 2. Kleefsman, K.M.T., Fekken, G., Veldman, A.E.P., Iwanowski, B., Buchner B.: A volume of fluid based simulation method for wave impact problems. J. Comp. Phys. 206(1), 363–393 (2005) 3. Luppes, R., Helder, J.A., Veldman, A.E.P.: Liquid sloshing in microgravity. Proceedings of the 56th International Astronautical Congress, IAC-05-A2.2.07. (2005) 4. Luppes, R., Helder, J.A., Veldman, A.E.P.: The numerical simulation of liquid sloshing in microgravity. Computational Fluid Dynamics 2006: ICCFD4, pp. 607–612. Springer, Berlin (2009). doi: 10.1007/978-3-540-92779-2 95 5. Luppes, R., Wemmenhove, R., Veldman, A.E.P., Bunnik, T.: Compressible two-phase flow in sloshing tanks. Proceedings of the 5th European Congress: ECCOMAS (2008) 6. Luppes, R., Wellens, P.R., Veldman, A.E.P., Bunnik, T.: CFD simulations of wave run-up against a semi-submersible with absorbing boundary conditions. Proceedings of the International Conference on Computational Methods in Marine Engineering. MARINE (2009) 7. Veldman, A.E.P.: The simulation of violent free-surface dynamics at sea and in space. Proceedings of the European Congress: ECCOMAS CFD, ISBN 909020970-0, paper 492 (2006) 8. Veldman, A.E.P., Gerrits, J., Luppes, R., Helder, J.A., Vreeburg, J.P.B.: The numerical simulation of liquid sloshing on board spacecraft. J. Comp. Phys. 224, 82–99 (2007) 9. Verstappen, R.W.C.P., Veldman, A.E.P.: Symmetry-preserving discretisation of turbulent flow. J. Comp. Phys. 187, 343–368 (2003) 10. Wemmenhove, R., Luppes, R., Veldman, A.E.P., Bunnik, T.: Application of a VOF method to model compressible two-phase flow in sloshing tanks. Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering: OMAE2008, paper OMAE200857254 (2008)

High-Precision Reconstruction of Gas-Liquid Interface in PLIC-VOF Framework on Unstructured Mesh Kei Ito, Tomoaki Kunugi, and Hiroyuki Ohshima

Abstract The authors are developing a high-precision numerical simulation algorithm for gas-liquid two-phase flows to simulate gas entrainment phenomena in fast reactors. In this study, the calculation method for the unit vector normal to a gas-liquid interface in the high-precision volume-of-fluid algorithm (PLIC-VOF) is improved on two-dimensional unstructured meshes. The improved calculation method utilizes the least-square-fit of the height function defined at adjacent interfacial cells. In addition, to conserve the gas and liquid volume in the interfacial cell, the geometric relationship of a reconstructed interface and the cell is formulated. As the result of a basic verification, the improved calculation method succeeds in reproducing rectilinear interfaces on an unstructured triangular mesh. In addition, it is confirmed that the improved calculation method enhances the simulation accuracy of the well-known slotted-disk revolution problem on an unstructured mesh.

1 Introduction In recent years, the PLIC (Piecewise Linear Interface Calculation)-VOF (Volumeof-fluid) algorithm [6] is often employed to accurately simulate gas-liquid twophase flows. The authors also employ the PLIC-VOF algorithm to simulate the gas entrainment (GE) phenomena in the large-sized sodium-cooled fast reactors in Japan (JSFR). In addition, we are developing the PLIC-VOF algorithm on unstructured meshes which are suitable for the numerical simulations of two-phase flows in highly complicated geometries like JSFR. First, the high-precision calculation methods were newly developed to improve the simulation accuracy. One of them is a volume-conservative advection method for the volume fraction, which is very efficient to improve simulation accuracy on highly-distorted unstructured meshes. Then, physics-basis formulations were also developed and employed to consider the appropriate mechanical balance near gas-liquid interfaces, e.g. the balance between K. Ito (B) Japan Atomic Energy Agency, Ibaraki 311-1393, Japan e-mail: [email protected]


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pressure and surface tension [1]. The developed PLIC-VOF algorithm was already applied to the numerical simulation of the GE phenomena in a simple experiment. As a result, the GE phenomena were well reproduced by the developed algorithm [2]. In this study, the calculation method for the unit vector normal to a gas-liquid interface (interfacial normal) is improved and verified on two-dimensional unstructured meshes.

2 Improved Calculation Method for Interfacial Normal It was confirmed that our PLIC-VOF algorithm is accurate enough on unstructured meshes compared to other famous algorithms. However, the developed algorithm is first-order because not all rectilinear interfaces are reproduced. In other words, the developed PLIC-VOF algorithm does not satisfy the requirement to be second-order [3]. Therefore, in this study, the calculation method of an interfacial normal in the PLIC-VOF framework is improved to meet the requirement. The procedure of the improved calculation method is as follows: (1) in an interfacial cell (central cell), an interfacial normal (np ) is predicted by the conventional calculation method, i.e. the Gauss–Green (G–G) method [5]; (2) in the adjacent interfacial cells to the central cell, the piecewise linear interfaces normal to the predicted interfacial normal are reconstructed; (3) the height function (H k in the interfacial cell k) is determined by Eq. (1) in all interfacial cells (the central cell and all adjacent interfacial cells) as the distance from a base-line in the predicted interfacial normal direction is: (1) H k = ck − cb · np , where ck and cb denote the definition points of the height function (the central point of the reconstructed interface) in each interfacial cell (k) and a point on the baseline, respectively; (4) in each interfacial cell, geometrical relationships are formulated as Eq. (2) to conserve gas and liquid volumes when the reconstructed interfaces are rotated (see Fig. 1):

Fig. 1 Geometric relationship in interfacial cell. The segment AB is the reconstructed interface, the segment D E is rotated interface with the angle of θ and the dashed segment DE is the corrected interface (parallel to D E ) to conserve gas and liquid volumes

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3 Verifications of Improved Calculation Method To verify the improved calculation method for an interfacial normal, two simple simulations are performed in this study. First, as a basic verification, the reproducibility of rectilinear interfaces is checked. The simulations are conducted in a square domain subdivided by unstructured triangular cells (see Fig. 2). Here, the rectilinear interfaces with angles of 0, 45, 90 and 135 degrees from the horizontal plane are located to pass through the domain center. Then, the precise volume fraction in each cell is calculated for each rectilinear interface and used to calculate the interface normal. The simulation results are summarized in Table 1. It is evident that the improved calculation method succeeds in reproducing the correct interfacial normal and the conventional G–G method fails.

Fig. 2 Unstructured triangular mesh

Table 1 Numerical errors in interfacial normal Correct value (deg.) Algorithm Average error Maximum error 0 45 90 135

G–G Improved G–G Improved G–G Improved G–G Improved

3.50 × 100 1.22 × 10−14 5.30 × 100 1.22 × 10−14 5.01 × 100 0.00 1.00 × 101 0.00

1.00 × 100 1.85 × 10−14 1.72 × 101 8.53 × 10−14 9.49 × 100 0.00 1.96 × 101 0.00

In addition to the basic verification, the well-known slotted-disk revolution problem [7] is solved to confirm the improved simulation accuracy on unstructured meshes. In this example, the simulation conditions are the same as Rudman’s [4] but a highly-distorted unstructured triangular mesh is employed. Then, a numerical error is estimated as the discrepancy of the volume fraction distribution before and after the one-full revolution around the domain center. As a result, the numerical error by the improved calculation method (9.50×10−3 ) is highly reduced compared

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to the error by the conventional G–G method (1.19 × 10−2 ) on the same unstructured mesh. It is amazing that the reduced error is less than the error by the G–G method on a structured mesh (1.07 × 10−2 ) with the same number of cells as the unstructured mesh.

4 Conclusion In this study, the improved calculation method of an interfacial normal in the PLICVOF framework is developed on unstructured meshes, and the simulation accuracy is highly enhanced by the improved calculation method.

References 1. Ito, K., Kunugi, T.: Appropriate formulations for velocity and pressure calculations at gasliquid interface with collocated variable arrangement. J. Fluid Sci. Technol. 4, 711–722 (2009) 2. Ito, K., Kunugi, T., Ohshima, H., Kawamura, T.: Formulations and validations of a highprecision volume-of-fluid algorithm on non-orthogonal meshes for numerical simulations of gas entrainment phenomena. J. Nucl. Sci. Tech. 46, 366–373 (2009) 3. Pilliod, J.E., Puckett, E.G.: Second-order accurate volume-of-fluid algorithms for tracking material interfaces. J. Comput. Phys. 199, 465–502 (2004) 4. Rudman, M.: Volume-tracking methods for interfacial flow calculations. Int. J. Numer. Meth. Fluids 24, 671–691 (1997) 5. Shahbazi, K., Paraschivoiu, M., Mostaghimi, J.: Second order accurate volume tracking based on remapping for triangular meshes. J. Comput. Phys. 188, 100–122 (2003) 6. Young, D.L.: Time-dependent multi-material flow with large fluid distortion. In: Morton, K.W., Baine, M.J. (eds.) Numerical methods for fluid dynamics, pp. 273–468. Academic Press, New York (1982) 7. Zalesak, S.T.: Fully multidimensional flux-corrected transport algorithm for fluids. J. Comput. Phys. 31, 335–362 (1979)

Part XXI

Mesh Adaptation

Adjoint-Based Methodology for Anisotropic Grid Adaptation Nail K. Yamaleev, Boris Diskin, and Kedar Pathak

Abstract A new adjoint-based grid adaptation methodology for solving steady and unsteady problems is presented. In contrast to conventional grid adaptation techniques utilizing the error equidistribution principle, the new approach directly solves an error minimization problem for which grid node coordinates are used as control variables. A minimum of the error functional is found using a gradient method based on the adjoint formulation. The mesh sensitivity derivatives required for solving the error minimization problem are computed using the solution of the corresponding adjoint equations. The key advantage of this formulation is that the adjoint equations are solved only once at each grid adaptation iteration regardless of the number of grid nodes, which makes this approach well suited for mesh adaptation. The new adjoint-based grid adaptation strategy is tested on several benchmark problems governed by the Euler and Poisson equations.

1 Error Minimization Problem The proposed anisotropic grid adaptation methodology is based on minimization of the error in a functional output (e.g. lift, drag, torque, thrust, etc.) over each time interval. The problem is to find a grid at each time step, which minimizes the error in the integral output of interest, while maintaining the same grid connectivity at all time levels. This time-dependent discrete error minimization problem can be formulated as follows: min en (Qn , Xn ), n X

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grid topology. The optimal control problem (1) is subject to the discretized flow equations. Along with the flow equations, a grid quality constraint is imposed to guarantee that no mesh degeneration occurs during the adaptation, which implies that there are no negative control volumes in the computational domain. To close the error minimization problem, the error functional in Eq. (1) has to be defined. Construction of reliable and accurate error estimates is a challenging problem, which is beyond the scope of the present paper. In the present analysis, the error functional is defined using the true error in an integral output en = ( fex − f )2 ,

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where f ex and f are the exact and computed integrals of a quantity of interest (e.g. pressure). The integral output is tolerant to non-smoothness of the flow variables and can be used for flows with strong discontinuities.

2 Adjoint-Based Anisotropic Grid Adaptation Method Finding a global extremum of the error minimization problem (1) is a difficult task. On one hand, the global minimum of this PDE-constrained minimization problem may not be feasible. On the other hand, for large-scale aerodynamic problems, the number of grid points and consequently the dimensionality of the design space may be of the order of O(107 ), thus making the computational cost associated with solution of this global minimization problem comparable or even higher than that required for achieving the same error level by uniform grid refinement. The above considerations suggest that instead of seeking the global minimum, it would be more practical to reduce the error by finding a local minimum of the error functional. A local extremum of the error minimization problem (1) can be efficiently found by gradient methods based on the adjoint formulation. Along with the efficiency, another key advantage of the adjoint-based method is that its computational cost is independent of the number of control variables (the grid node coordinates), which makes this approach particularly attractive for mesh adaptation. The discrete PDE-constrained error minimization problem (1) is solved by the method of Lagrange multipliers which is used to enforce the governing equations and the corresponding boundary conditions as constraints. The discrete Lagrangian functional is defined as follows:

T 3 n Qn −Qn−1 1 n−2 Qn−2 −Qn−1 n + Rn n−1 V + V + R Q L(Q, X, Λ) = en + Λn GC L 2 Δt 2 Δt (3) where Λ is a vector of Lagrange multipliers, en is an error functional, Rn and RnGC L are the flow and geometric conservation law residuals, respectively. To compute the mesh sensitivity derivatives, the Lagrangian is differentiated with respect to Xn . This leads to the following adjoint equations for determining the Lagrange multipliers (see [2] for details):

Adjoint-Based Methodology for Anisotropic Grid Adaptation 3 n n 2Δt V Λ

+

∂Rn ∂Qn

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The main advantage of the adjoint formulation is that at each optimization iteration, the adjoint equations (4) should be solved once regardless of the number of grid points. The above adjoint equation involves only the adjoint variable at the current time level, thus making it similar to that of the steady state adjoint formulation in the sense that no integration of the adjoint equations backward in time is required. With the Lagrange multiplier Λn found from Eq. (4), the sensitivity derivative is evaluated as follows: dL ∂en = + [Λn ]T dXn ∂Xn

∂RnGC L n−1 3 ∂Vn Qn − Qn−1 ∂Rn + Q + 2 ∂Xn Δt ∂Xn ∂Xn

(5)

A local minimum of the error functional at each time step is found by the steepest descent method in which each step of the optimization cycle is taken in the negative gradient direction Xnk+1 = Xnk − δk

dL dXnk

T ,

(6)

where δk is an optimization step size which is chosen adaptively [4], k is the optimization iteration number. The iteration process is repeated until either the error in the integral output becomes smaller than a user-specified error tolerance or the difference between the corresponding grid node coordinates on the previous and updated meshes becomes less than a given threshold value. To avoid overlapping of neighboring control volumes during adaptive r refinement given by Eq. (6), a trust region approach is used. At each optimization iteration, a trust region for the local steepest descent search algorithm is defined such that each node remains within its control volume. Since overlapping of grid cells is not allowed, the number of grid points and grid topology remain unchanged, as follows from Eq. (6). This adaptive r-refinement procedure redistributes grid nodes in anisotropic fashion to minimize the error estimate. In contrast to heuristic approaches relying on the Hessian of a single flow variable to determine stretching and orientation of mesh cells during grid adaptation [3], the proposed adjoint-based r-refinement technique (4), (5) and (6) provides a rigorous relation between the sensitivity of the integral output error and the size, shape, and orientation of mesh elements.

3 Numerical Results The first test case is presented to evaluate the performance of the proposed grid adaptation strategy for the 2-D Euler equations describing the steady shocked flow in a diverging nozzle. The nozzle is symmetric about the x-axis, and its cross-sectional

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area is given by the following equation: A(x) = 1.398 + 0.347 tanh(0.8x − 4), 0 ≤ x ≤ 10. The inflow Mach number is set equal to 1.2, and the outflow pressure has been chosen to be 1.073847, so that the flow is transonic and the shock wave is located near x = 3. The functional considered for this test case is the pressure integral computed along the nozzle walls. The error in the functional output is estimated using a numerical solution obtained with the 4th-order finite difference method on a fine 961 × 97 mesh. The initial mesh is constructed from a smooth structured quadrilateral 81 × 9 mesh by dividing each quadrilateral element in two triangles in such a way that the mesh is symmetric with respect to the x-axis. To assess the performance of the dynamic variant of the new adjoint-based grid adaptation procedure for unsteady flows, the steady transonic nozzle flow problem is solved in a time-dependent fashion. The 2-D Euler equations are integrated in time with a constant nondimensional time step that is set equal to 20. Sixteen time steps are required to reduce the residual below 10−12 . At each time step, the grid adaptation iterations are repeated until the objective functional becomes smaller than a user-defined tolerance which is set equal to 10−12 . Only 3–6 iterations are needed to reach convergence at each time level. Time histories of the error in the pressure integral and the mesh sensitivities are presented in Fig. 1. The objective functional converges to zero, as the mesh is adapted. The optimal mesh obtained at the final time level is depicted in Fig. 2. Note that this test problem can be solved using the static variant of the adjoint-based grid adaptation algorithm which results in a different optimal mesh that also nullifies the objective functional. This observation suggests that there exist multiple solutions to the error minimization problem (1) and (2), and the adjoint-based steepest descent method finds an optimal mesh that is close to the initial mesh. The second test problem is a 2-D scalar Poisson equation with a manufactured solution resembling the boundary-layer profile. The equation is discretized with a second-order node-centered finite volume scheme [1]. The right hand side and

Fig. 1 Time histories of the pressure integral error (left) and the L ∞ norm of the mesh sensitivity derivatives (right) obtained for the nozzle flow problem

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the Dirichlet boundary conditions for the Poisson equation are chosen so that the problem has the following exact solution: q(x, y) = (x + 1) tanh(25y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 with a thin boundary layer along the lower boundary y = 0. The integral of the 51 normal derivative of q along the boundary y = 0, f (q) = ∂q ∂ y d x, is used as 0

the quantity of interest. The static formulation of the adjoint-based grid adaptation algorithm is used. At each optimization iteration, the residuals of the discretized Poisson equation and its adjoint are driven to the machine zero. Convergence histories of the error functional and the maximum norm of the mesh sensitivity derivatives are presented in Fig. 3. The error in the target integral and the mesh sensitivities drop by more than three orders of magnitude every 10 optimization iterations and become less than 10−9 after 30 iterations, thus indicating that the optimizer converges to a global minimum of the error minimization problem. The initial mesh and the final adaptive mesh obtained at the 33rd optimization iter-

Fig. 3 Convergence of the integral error (left) and the L ∞ norm of the mesh sensitivity derivatives (right) for the Poisson problem

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ation are shown in Fig. 4. Despite that the initial mesh is clustered near the lower boundary, the functional error on this mesh is of the order of one. The comparison of the initial and adaptive grids shows that small changes in the grid node coordinates result in the drastic error reduction in the integral of the normal derivative. Similar to the previous test case, the error minimization problem subjected to the Poisson equation has multiple solutions that can readily be obtained by starting from different initial meshes. It again indicates that a solution of the error minimization problem is not unique.

References 1. Anderson, W.K., Bonhaus D.L.: An implicit upwind algorithm for computing turbulent flows on unstructured grids. Comput. Fluids 23, 1–21 (1994) 2. Nielsen, E.J., Diskin B., Yamaleev, N.K.: Discrete adjoint-based design optimization of unsteady turbulent flows on dynamic unstructured grids. AIAA J. 48(6), 1195–1206 (2010) 3. Perarie, J., Vahdati, M., Morgan, K., Zenkiewicz, O.C.: Adaptive remeshing for compressible flow computations. J. Comput. Phys. 72, 449–466 (1987) 4. Yamaleev, N.K., Diskin B., Nielsen, E.J.: Adjoint-based methodology for time dependent optimization. AIAA Paper 2008–5857 (2008)

Transient Adaptive Algorithm Based on Residual Error Estimation N. Ganesh and N. Balakrishnan

Abstract In this paper, we propose a novel algorithm for transient mesh adaptation based on residual error estimation. The error estimator, known as the $-parameter is a reliable measure of the local truncation error and is used to derive local length scales that guide refinement and derefinement. In order to efficiently handle moving fronts, a Refinement Level Projection (RLP) algorithm is developed, which guarantees that the flow features remain within refined zones at all times. Dynamic adaptation of inviscid transonic flow past a pitching airfoil illustrates the efficiency of the proposed algorithm.

1 Introduction Several engineering problems encountered in practice involve unsteady flow phenomena such as periodic vortex shedding and shock propagation among others. Obtaining high resolution solutions for such time-dependent problems would require a uniformly fine mesh in the entire domain if adaptive algorithms are not employed. Such an approach is obviously not cost-effective and becomes very expensive for practical 3-D applications. This motivates the development of an efficient dynamic adaptive algorithm for accurate simulations of transient phenomena.

2 Transient Adaptive Algorithm The ideal transient adaptation algorithm consists of two steps: (1) Obtaining the “best” mesh for a given time level and (2) Predicting the evolution of flow phenomena and refining regions through which the phenomena progresses. The first step results in the best spatial distribution of volumes that resolves the flow features at a given time level. As the time progresses, flow phenomena on the “best” mesh get convected out from refined zones into unrefined regions resulting in a diffusive N. Ganesh (B) St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN, USA e-mail: [email protected]


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solution and loss in accuracy. Ideally, one would adapt the mesh at every time level, but such a procedure is computationally expensive and can introduce interpolation errors during solution transfer between meshes. It is therefore desirable to adapt the mesh with a finite frequency (once in a few timesteps), to evolve a computationally efficient transient adaptive algorithm with lesser interpolation errors. However, such an approach must also guarantee uniform accuracy of the solution by ensuring that the flow features remain in adapted regions between successive adaptations. This is achieved in the second step, thanks to a simple strategy known as “Refinement Level Projection” (RLP). In this strategy, the refinement levels of cells on the “best mesh” are projected over a period of time which corresponds to the time interval between successive adaptations. The projection of refinement levels involves the definition of a “projection radius” which depends on some estimate of the velocity of the flow features, the physical time step and the time interval between successive adaptations. Effectively, the RLP strategy constructs a “buffer” of cells that guarantee that the flow features (which are also the error sources) remain within adapted zones throughout and keeps the spatial error levels within acceptable bounds. The two–step transient adaptive algorithm can be summarised below. Let G and S represent the grid and the solution respectively. Let i denote the iteration count in obtaining the “best” mesh and superscripts n, n − 1 and n − 2 denote the current time level and the two earlier time levels respectively. We shall refer to S˜ as solutions obtained on a given grid by mapping from a previous level mesh. The physical time step is denoted as Δt. The velocity of the flow feature is denoted as Vfeature while Rmax denotes the maximum number of refinement levels. The number of iterations between two successive adaptations is defined as the adaptation frequency and is denoted as N f . STEP 1: Determine the “best” mesh at t n given the mesh–solution pair G n0 , S0n , Rmax and Δt. 1. Based on the solution Sin on the mesh G in , flag the cells based on a suitable adaptation strategy and adapt the mesh. Refinement of the mesh is based on error indicators and the residual error estimator, while derefinement is effected purely using the error estimator. The details of residual estimation for unsteady flows is n . discussed in Sect. 3. This gives the adapted mesh G i+1 n−1 n−2 and Si available on the grid G in to the 2. Transfer (or Map) the solutions Si n adapted grid G i+1 using a constant polynomial interpolation strategy. Let the n−1 n−2 n be represented as S˜i+1 and S˜i+1 . mapped solutions on the grid G i+1 n 3. Solve the transient flow problem on mesh G i+1 for a single physical time step n on corresponding to the time interval [t n − Δt,t n ]. This gives the solution Si+1 n the adapted mesh at time level t . 4. Set i = i + 1 and repeat 1–3 until termination, the termination criterion being i > Rmax . This give the “best” mesh at time level t n , denoted as Gˆ n and the corresponding solution Sˆ n .

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STEP 2: Refinement Level Projection (RLP), given the mesh-solution pair (Gˆ n , and N f

Sˆ n )

1. Define the set of cells on Gˆ n whose refinement levels are to be projected. 2. For every cell in this set, define a radius of projection, Δr proj = η × Vfeature × N f × Δt, where η is a constant to account for the uncertainty in the estimate of feature velocity. 3. For every cell in the set and whose level is l, refine all cells which have a level less than l and that fall within its projection radius, until the cell receives the level l. 4. Transfer (or Map) the solution Sˆ n from Gˆ n onto the new mesh obtained after projection. This give the final adapted mesh after projection, G n and the corresponding solution S n . The transient flow problem is now solved on the grid G n over the time interval [t n , t n + N f Δt], after which the transient adaptation is again enforced. For more details, refer [1].

3 Error Estimation for Unsteady Flows In this section, we present the theory of residual error estimation for unsteady flows. Consider the conservation law given by I [U ] = 0. Splitting the exact spatio-temporal operator I into its constituent spatial and temporal operators, represented by I h and I t respectively, we have, I t [U ] + I h [U ] = 0

(1)

A numerical solution u to this conservation equation demands a suitable discretisation and the discrete approximation to Eq. (1) reads, δ 1 [u] = δ 1,t [u] + δ 1,h [u] = 0

(2)

where δ 1,t and δ 1,h are the temporal and spatial operators constituting δ 1 . The residual estimator is the imbalance arising from the use of the exact operator with the numerical solution. This immediately leads to, I t [u] + I h [u] = R 1,t [u] + R 1,h [u] = R 1 [u]

(3)

Our interest lies in estimating only the spatial component of the truncation error and therefore employing the same discrete temporal operator δ 1,t and a different spatial operator δ 2,h as discrete approximations to I t and I h in Eq. (3), leads to,

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where R 2,h [u] is the error due to the discretisation of the exact spatial operator I h . Using Eq. (2) in Eq. (4), yields the final expression for the residual error estimator as, δ 2,h [u] − δ 1,h [u] = R 1,h [u] − R 2,h [u] = $[u].

(5)

It is evident that if R 1,h [u] ∼ O(hm ) and R 2,h [u] ∼ O(hn ) and m < n, then the $-parameter will be an estimate of the local truncation error associated with the spatial discretisation. The definition of $-parameter given by Eq. (5) can be effectively employed even in a steady state problem, to account for the non-convergence of the numerical problem to steady state. In that sense, error estimation for unsteady flows is a generalisation of the error estimation procedure for steady flows proposed by the authors in [2–4]. To compute the $-parameter, δ1,h u is calculated by integrating the fluxes on each interface with a single-point Gaussian quadrature, with the states obtained using a linear reconstruction while δ 2,h u is calculated in a similar fashion but with three – point quadrature and quadratic reconstruction. For more details, please refer [1].

4 Results In order to employ the residual error estimator in the transient adaptive algorithm, it is necessary to establish the estimator to consistently estimate the truncation error. We consider, for this purpose the isentropic vortex convection in an inviscid flow [1]. The initial condition is given by the exact solution at time t = 0 and extrapolation boundary conditions are employed. The computational domain is [−50, 50] × [−5, 5] and the vortex is initially centered at the origin. The unsteady HIFUN-2D solver [1] is used for all computations, which are performed upto t = 0.01 using a physical time step of 0.001. Experiments on moving structured and unstructured meshes show that the $-parameter indeed decreases with grid refinement, with a rate close to 2 and 1 respectively. Subsequently, we attempt to solve the inviscid transonic flow past a pitching NACA0012 airfoil using the transient adaptive algorithm based on the residual estimator. The sinusoidal pitch motion is defined by, α(t) = 0.016 + 2.51 sin(0.1628t) and the mach number is 0.755. A circular computational domain is chosen with a farfield of 10 chords and the entire grid is rigidly moved. Empirical studies suggest a feature velocity of 0.05 for this case. The physical time step of 0.01 and adaptation is performed once in every 100 time steps for 40 adaptation cycles. Projection uses an η value of 1.3 and maximum three levels of refinement is considered. As the grid is moved rigidly and the slipstream moves passively with the grid, there is no refinement level projection required for the cells which are detected by the curl of velocity. The adapted mesh and mach contours at two different time instants

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(Figs. 1 and 2) as well as the pressure distribution over the airfoil (Figs. 3 and 4) show excellent resolution of the shocks and establishes the efficacy of the proposed algorithm.

5 Conclusions A new residual error estimator for unsteady flows and an associated novel transient adaptive algorithm are proposed. Dynamic adaptation of the flow past a pitching airfoil demonstrates the efficacy of the proposed algorithm.

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Fig. 3 Mesh and mach contours at α = 1.67◦ (↑). (a) Adapted Mesh. (b) Mach contours

Fig. 4 C p distribution for α = 2.01◦ (↓) and α = −0.54◦ (↑)

References 1. Ganesh, N.: Residual error estimation and adaptive algorithms for compressible flows. Ph.D. Thesis. Department of Aerospace Engineering, Bangalore, India (2010) 2. Ganesh, N., Balakrishnan, N.: An adaptive strategy based on local truncation error. Proceedings of Eight Asian Computational Fluid Dynamics Conference (ACFD8), HongKong, 2010 3. Ganesh, N., Shende, N.V., Balakrishnan, N.: A residual estimator based adaptation strategy for compressible flows. In: Deconock, H., Dick, E. (eds.) Computational Fluid Dynamics 2006, pp. 383–388 (2009) 4. Ganesh, N., Shende, N.V., Balakrishnan, N.: $-parameter: A local truncation error based adaptive framework for finite volume compressible flow solvers. Comput. Fluids 38, 1799–1822 (2009)

Adaptive and Consistent Properties Reconstruction for Complex Fluids Computation Guoping Xia, Chenzhou Lian, and Charles L. Merkle

Abstract An efficient reconstruction procedure on adaptive Cartesian mesh for evaluating the constitutive properties of a complex fluid from general or specialized thermodynamic databases is presented. Reconstruction is accomplished on a triangular subdivision of the 2D Cartesian mesh covering thermodynamic plane of interest that ensures function continuity across cell boundaries to C 0 , C 1 or C 2 levels. The C 0 and C 1 reconstructions fit the equation of state and enthalpy relations separately, while the C 2 reconstruction fits the Helmholtz or Gibbs function enabling EOS/enthalpy consistency also. All three reconstruction levels appear effective for CFD. The time required for evaluations is approximately two orders of magnitude faster with the reconstruction procedure than with the complete thermodynamic equations. Storage requirements are modest for today’s computers, with the C 1 method requiring slightly less storage than those for the C 0 and C 2 reconstructions when the same accuracy is specified. Sample fluid dynamic calculations based upon the procedure show that the C 1 and C 2 methods are approximately a factor of two slower than the C 0 method but that the reconstruction procedure enables arbitrary fluid CFD calculations that are as efficient as those for a perfect gas or an incompressible fluid for all three accuracy levels.

1 Introduction Computational fluid dynamics requires the coupled solution of the partial differential equations that comprise the basic conservation laws (mass, momentum and energy) in combination with an auxiliary set of constitutive relations in order to close the system. Simple algebraic relations are enough for perfect gases and incompressible fluids. There are, however, many engineering applications in which gases or vapors do not follow the perfect gas laws and liquids can not be treated as incompressible. For such applications, it is necessary to turn to more general equations of state. G. Xia (B) Purdue University, West Lafayette, IN 47907, USA, e-mail: [email protected]


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One approach is to use the family of classical semi-theoretical expressions that include the van der Waals, Peng–Robinson and Redlich–Kwong–Soave (RKS) equations of state in which the perfect gas relation is modified by adding additional terms and constants to improve accuracy in regions where real gas effects become important. Although these methods provide pressure-temperature-density relations that are accurate over a broader range, they generally do not provide an analogous expression for the internal energy and relations analogous to those for perfect gases are often used for the specific heats. A second alternative is to use more complete and accurate thermodynamic databases such as REFPROP [2], which provide highly accurate properties across the liquid-vapor-supercritical regimes for a large variety of fluids including full pressure-density-temperature-internal energy relations. In using any of these equations of state, a variety of difficulties can be encountered that increases the cost and complexity of a CFD solution. In the present paper we develop a generalized properties evaluation procedure based upon an adaptive table look-up method that provides flexibility, efficiency and accuracy for nearly any conceivable property formulation. The flexibility comes because the equation of state is inverted by separate software outside the CFD code thereby enabling the independent variables pair to be chosen by the user while simultaneously enabling the entire variety of equations of state to be incorporated in a CFD code in a common fashion. The efficiency comes from using a tree-based data structure that provides fast table look-up over large or small thermodynamic domains. The accuracy comes from an adaptive Cartesian mesh that adjusts the table resolution to user specified input criteria and evaluates both the thermodynamic functions and their derivatives. Although the procedure is applicable to any properties formulation, we base our examples on information obtained from REFPROP.

2 The Conservation Equations and Fluid Properties The conservation form of the mass, momentum and energy equations for an arbitrary, Newtonian fluid are [1]: ∂Vi ∂Q ∂E i ∂Q ∂Q p = +H + + ∂Q p ∂τ ∂t ∂xi ∂xi

(1)

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variables pair along with the velocity components and the mass fractions. The Jacobian matrix that multiplies the pseudo-time term in Eq. (1) and the product matrix from which the eigenvalues of the system are: ⎛ ⎞ ρp 0 ρT ∂Q ⎠ and ρ p u ρT u =⎝ ρ ∂Q p o o ρ p h + 1 − ρh p ρu ρ p h + ρh T J ⎛ ⎞ 0 ρh T Ju ∂Q p ∂E u J 0⎠ Ap = = ⎝1 ρ ∂Q ∂Q p 0 1 − ρh p u

(3)

where = ρ p h T + ρT (1 − ρh p )/ρ. Note that both matrices contain the density plus four thermodynamic derivatives: ρ p , ρT , h p and h T . These derivatives directly affect the fluids solution and must be specified from the fluids database. Note that these four thermodynamic derivatives, ρ p , ρT , h p and h T , can be evaluated in terms of the first and second partial derivatives of the Gibbs function: ρp = −

gpp gpT , ρT = − 2 , h p = g p − TgpT , and h T = −TgTT 2 gp gp

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For compactness of notation, we have given these in terms of the onedimensional equations with only a single species. In mixture computing, the above thermodynamic property of the mixture can be evaluated from the Amagat’s law.

3 Adaptive Cartesian Grid Methods and Consistent Reconstruction In the present paper we choose a tree-based, adaptive Cartesian grid method that enables both fast search and local refinement for property storage. In any p-T plane (or other thermodynamic plane [3]), we begin by defining a rectangular region that includes the physical domain of interest. The requisite properties are then evaluated at each of the four corners of the square and an appropriate reconstruction is used to evaluate these properties at pre-selected points in the four quadrants of the square. The reconstructed properties are then compared with their “exact” values obtained from the complete property equations and used to assess the reconstruction error. If the error in any square exceeds a user-specified threshold, that square is subdivided into four smaller squares. Three different levels of reconstruction are considered, involving C 0 , C 1 , and C 2 continuity across cell boundaries whose results range from partially to fully consistent. In the C 0 method, values for each of the six thermodynamic fluid properties are stored at the vertices of each square in the mapped plane. The squares are then subdivided into triangles and the values of each function at the three vertices are used to construct independent bi-linear reconstruction functions for each property.

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Subdividing the squares ensures continuity of the function across the interface of squares of different sizes. The C 1 method treats the density and enthalpy as unrelated functions that are each handled in analogous fashion. Using the density as an example, the goal of C 1 reconstruction is to reconstruct the density function in such a manner that the values of ρ, ρ p , and ρT are consistent within any given cell and that all three quantities are continuous across cell boundaries. This level of consistency and continuity can be achieved by using a bi-quintic (fifth-order) polynomial. The C 2 method interpolates all properties and their derivatives by reconstructing the Gibbs function as a ninth-order polynomial.

4 Results Figure 1 shows the grid structure produced for CO2 . It comprises the temperature and pressure ranges, 500 K ≤ T ≤ 1,600 K, 0.01 MPa ≤ p ≤ 1,000 MPa and lies entirely within the vapor region. The grid of this zone is for a specified accuracy of 1% with C 0 continuity. The grid color is keyed to the magnitude of the density. From the plot in Fig. 1 more refinement around the critical point (top left corner) is clearly observed. In the lower pressure portion, CO2 behaves as a perfect gas and thus less grid points are used. The 1% accuracy case requires a total of ten levels of refinement and results in 22,000 reconstruction points. A further refinement to 0.1% accuracy (not shown here) requires 11 refinement levels and 225,000 points. The comparison 3

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of the table size and timing are summarized in Table 1. Evaluation timing is based on the time consumption of performing 100,000 property calculations along the two diagonal dashed lines in Fig. 1. The evaluation times in column 4 of Table 1 are only slightly longer than that for a table of uniform grid size, which is considered advantageous on searching. However, the adaptive table uses only a fraction of the storage of the uniform table in order to achieve the same accuracy. Taking the 1% accuracy table as an example, a uniform table refined to the same accuracy as the adaptive one will have 1,048,576 cells, 50 times larger than that in the adaptive table. The table look-up and evaluation time is essentially independent of the table size. Direct calculation from the property database, REFPROP, is more than two orders slower than the adaptive table look-up procedure, as shown in Table 1.

Fluid CO2 H2 O

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equal-sized mesh (a uniform mesh refined to the deepest level).

The table sizes required for the C 0 , C 1 and C 2 reconstruction procedures is compared in Fig. 2 from both a real fluid property from REFPROP and perfect gas property. In general C 1 and C 2 methods need fewer grid points due to the increased accuracy of the interpolating functions, but they require the storage of more values (coefficients in the bi-variate polynomials) per grid point. A significant reduction in database size can be achieved when moving from C 0 reconstruction to its C 1 counterpart. Going from C 1 reconstruction to C 2 reconstruction will result in a modest increase of the database size. 1290

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A one-dimensional example of inviscid subsonic flow through a convergentdivergent nozzle is presented to demonstrate the method. The nozzle is symmetric about the throat with the inlet and exit areas 1.25 times the throat area. The working fluid is chosen as H2 O with an inlet total pressure of 60 MPa and stagnation temperature of 750 K. The back pressure at the outlet is 50 MPa. A total of 200 cells are used in the calculation. The convergence rates for this case using the C 0 , C 1 and C 2 methods are shown in Fig. 3. The convergence with all three of these reconstruction methods is identical to the convergence obtained by coupling the REFPROP routines directly into the CFD code. The convergence with discontinuous interpolation, however, stalls.

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Fig. 3 Exemplary plot showing effect of triangulation on convergence of a specific CFD calculation

Although the number of iterations is essentially identical for the different property evaluation methods, their CPU costs are significantly different. Figure 4 shows the cumulative CPU time required for the computation as a function of iteration number for 1,000 iterations. The C 0 reconstruction method results in the fastest execution. Comparison of the results for the inconsistent and the consistent interpolation methods indicates that the consistent interpolation method increases the CPU time by about 5%. The costs of the C 1 and C 2 methods are nearly equal and are approximately twice that of the C 0 method. The calculation based on the exact property evaluation from REFPROP is approximately 160 times slower than the C 0 method and 80 times slower than the C 1 and C 2 methods.


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The CPU time for a CFD calculation can be broken into two parts: the property evaluation time and the equation solution time. The relative advantages of faster property evaluation times clearly will decrease as the complexity of the equation solution increases, and in particular as we move from one to three dimensions. The nominal cost of the property evaluation by REFPROP in this region of the H2 O map is about 450 times that of the C 0 integration method (see Table 1). For this onedimensional case, the CFD calculation time ratio is 160, suggesting that the ratio of equation solution time to property evaluation time in a one-dimensional calculation is approximately 1.8. Numerical computations show that the one-dimensional solver requires about 61.25 μs/(cell × iteration). Corresponding costs for two- and threedimensional solutions are 100.2 and 200.9 μs/(cell×iteration) respectively. Assuming the property evaluation time does not change when going from one to three dimensions, this suggests that a two-dimensional calculation with C 0 interpolation method will be 115 times faster than the direct REFPROP evaluation calculation and the three-dimensional calculation will be 66 times faster. Solutions based upon C 1 and C 2 reconstruction will be about half this amount.

5 Summary and Conclusions An adaptive reconstruction method for fluid dynamics computations of complex fluids with general equations of state is presented. The technique is based upon a Cartesian-grid approach that uses a binary tree data structure to store property

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information in an unequally spaced table whose resolution is automatically chosen to provide user-specified accuracy. The efficiency and accuracy of the method are assessed by comparing with the thermodynamics properties obtained from the REFPROP database. The resulting property reconstructions are nominally two orders of magnitude faster than property evaluations from the original database. This properties evaluation advantage translates into solution time enhancements of approximately one to two orders of magnitude for 3-D CFD computations. Discontinuity in the reconstruction is removed to improve the convergence. Reconstruction of the interpolating functions at C 0 , C 1 and C 2 continuity is assessed and compared. The convergence rate in example calculations is identical for the three reconstruction methods and for direct property evaluations from REFPROP, but the CPU costs are approximately two orders of magnitude smaller. The CPU cost of the C 0 reconstruction method is approximately half that of the C 1 method, while the C 1 and C 2 methods are about the same because the C 1 method constructs two fifth-order polynomials as opposed to one ninth-order polynomial in the C 2 method.

References 1. Merkle, C.L., Sullivan, J.Y., Buelow, P.E.O., Venkateswaran, S.: Computational of flows with arbitrary equation of state. AIAA J. 36(4), 515–521 (1998) 2. NIST Standard Reference Database 23, NIST Thermodynamic properties of refrigerant mixtures database (REFPROP), Version 4.0, Gaithersburg, MD (1993) 3. Swesty, F.D.: Thermodynamically consistent interpolation for equation of state tables. J. Comput. Phys. 127, 118–127 (1996)

Space-Filling Curve Techniques for Parallel, Multiscale-Based Grid Adaptation: Concepts and Applications Sorana Melian, Kolja Brix, Siegfried Müller, and Gero Schieffer

Abstract The concept of fully adaptive multiscale finite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In order to perform 3D computations for complex geometries in reasonable CPU time, the underlying multiscale-based grid adaptation strategy has to be parallelized on distributed memory architectures. In view of a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be well-balanced and the communication between processors has to be minimized. This has been realized using space-filling curves. A 3D simulation of a Lamb–Oseen vortex is presented.

1 Introduction The numerical simulation of (compressible) fluid flow requires highly efficient numerical algorithms which allow for high resolution of all physical waves occurring in the flow field and their dynamical behavior. In order to use the computational resources (CPU time and memory) in an efficient way, adaptive schemes are wellsuited. By these schemes, the discretization is locally adapted to the variation of the flow field. The crucial point is the design of a criterion that allows to decide whether to locally refine or coarsen the grid. Here we use multiscale techniques that aim at data compression, see [5] for an overview. Although this multiscale-based grid adaptation concept leads to a significant reduction of the computational complexity in comparison to computations on uniform meshes, this is not sufficient to efficiently perform 3D computations for complex geometries. In addition, we need parallelization techniques in order to further reduce the computational time. For this purpose, a parallelized version of the S. Müller (B) Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, 52056 Aachen, Germany e-mail: [email protected]


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multiscale transformation library, developed in [4], has been embedded into the finite volume solver Quadflow, see [2]. Here we will briefly summarize the basic concepts and present an application to the simulation of a Lamb-Oseen vortex. More details can be found in [3].

2 Multiscale-Based Grid Adaptation Finite volume schemes have been frequently applied to the discretization of balance equations arising for example in continuum mechanics. In the past decade, a new adaptive concept for finite volume schemes has been developed. This is based on multiscale techniques, where by means of a multiscale analysis, associated increasing resolution, i.e., with a hierarchy of nested E grids Gl , l = 0, . . . , L, with E Gl := {Vλ }λ∈Il with λ∈Il Vλ = Ω ⊂ Rd and Vλ = μ∈M 0 ⊂Il+1 Vμ , λ ∈ Il , a λ locally refined grid can be constructed. For this purpose, we first perform a multiscale decomposition of the data on the finest resolution level L. These data are 5 determined by cell averages uˆ λ := |V1λ | Vλ ud V, λ ∈ Il of an integrable function u. They are decomposed into coarse scale information uˆ λ , λ ∈ Il , and details dλ , λ ∈ Jl , encoding the difference between two refinement levels. This is realized by successively applying the two-scale relations uˆ λ =

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In order to account for the dynamics of a flow field due to the time evolution and to appropriately resolve all physical effects on the new time level, this set is to be inflated such that the prediction set D˜ L ,ε ⊃ D L ,ε contains all significant details of the old and the new time level. In a last step, we construct the locally refined grid, see Fig. 1 (right), and the corresponding cell averages. For this purpose, we proceed levelwise from coarse to fine, see Fig. 1 (left), and check for all cells of a level whether there exists a significant detail. If one is found, then we refine the respective cell, i.e., we replace the average of this cell by the averages of its children by locally applying the inverse two-scale transformation

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uˆ μ =

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l,1 l,0 l,1 Note that the mask coefficients m l,0 μ,λ , m μ,λ and gμ,λ , gμ,λ in (1) and (3), corresponding to the index sets Mλ0 , Mλ1 , and G 0λ , G 1λ , respectively, do not depend on the data but on geometric information only. For specific examples E Lwe refer to [5]. ˜ L ,ε ⊂ The final grid is then characterized by the index set G l=0 Il such that E λ∈G˜ L ,ε Vλ = Ω. In order to proceed levelwise, we have to inflate the prediction set such that it corresponds to a graded tree. This also guarantees that there is at most one hanging node per cell face, see [4].

3 Parallelization Using Space-Filling Curves When it comes to parallelization, a data parallel approach is usually appropriate. On a distributed memory architecture, the performance of a parallelized code crucially depends on the load-balancing and the overhead induced by the interprocessor communication. As the starting point is a hierarchy of nested grids, we do not need a partition of a single uniform refined mesh, but a partition of a locally refined grid, where not all cells on all levels of refinement are active. A natural representation of a multilevel partition of a mesh is a global enumeration of the active cells. We are aiming at the design of a method to realize this at runtime, as the adaptive mesh is also created at runtime using the multiscale representation techniques. Such an enumeration is provided by a Space-Filling Curve (SFC), where the basic idea is to map level-dependent multiindices identifying the cells in a dyadic grid hierarchy to a one-dimensional curve, i.e. the unit square (2D) or the unit cube (3D) is mapped to the unit interval by rescaling. For our type of discretization, Hilbert SFCs are well suited, helping to reduce the overall computational time by achieving very good load-balance and in the meanwhile saving computational resources (time and

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Fig. 2 First 4 iterates of 2D Hilbert space-filling curve

memory). Figure 2 shows the first iterates of a 2D Hilbert SFC. For a detailed discussion on the construction on the Hilbert space-filling curve we refer the reader to [6, 8]. Thus, each of the cells of the adaptive grid has a corresponding unique number on the curve, due to the global enumeration. The unit interval is then split into different parts containing approximately the same number of entries. Each of these parts is mapped to a different processor, so that we obtain a well-balanced partition, but we also have to pay the cost of interprocessor communication, as neighbors from the geometrical domain may belong to different processors. For a more specific example, Fig. 3 shows a locally refined grid with three levels of refinement where each cell is mapped to a position on the Hilbert SFC and then a partition to three processors can be determined only by ordering the numbers along the curve and then distributing it to the desired number of processors. A well-balanced distribution of a locally refined grid to five processors can be seen in Fig. 4. For further details, we refer to [3].


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4 The Solver Quadflow The above multiscale-based grid adaptation concept has been integrated into the solver Quadflow [2]. This solver has been developed for more than one decade within the collaborative research center SFB 401 Modulation of Flow and FluidStructure Interaction at Airplane Wings, cf. [1, 7]. In order to exploit synergy effects, it has been designed as an integrated tool where each of the core ingredients, namely,

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(i) the flow solver concept based on a finite volume discretization, (ii) the grid adaptation concept based on wavelet techniques, and (iii) the grid generator based on B-spline mappings, is adapted to the needs of the others. In particular, the three tools are not just treated as independent black boxes communicating via interfaces. Instead, they are highly intertwined on a conceptual level mainly linking (i) the multiresolution-based grid adaption that reliably detects and resolves all physical relevant effects, and (ii) the B-spline grid generator which reduces grid changes to just moving few control points whose number is, in particular, independent of any local grid refinement. The mathematical concepts have been complemented recently by parallelization techniques that are indispensable for further reducing the computational time to an affordable order of magnitude when dealing with realistic 3D computations for complex geometries, cf. [3]. Note that the flow solver and the multiscale-based grid adaptation have totally different algorithmic requirements: on the one hand, there is a finite volume scheme working on arbitrary, unstructured discretizations; on the other hand, there is the multiscale algorithm assuming the existence of hierarchies of structured meshes. The flow solver module is face-centered, since the central item is the computation of the fluxes at the cell faces, while the adaptation module is cell-centered, analyzing and manipulating cell averages. Moreover, the data structures used in the two parts are also different: while for the adaptation part hash maps are used, for the flow solver part more simple data structures (i.e. arrays) are applied. The link between these two modules is done through a data conversion algorithm, which organizes all the data communication – the transfer of the conservative variables, volumes, cell centers, the registration of the nodes, the construction of the faces and determination of their neighboring cells and nodes – between the two modules is a connectivity list (see [2]). The parallelization of such an algorithm is not trivial. Due to the adaptivity of the mesh, the connectivity list might need to be reconstructed at any time in the computation, when also rebalancing and repartitioning is required. Thus the cells located at the partition’s boundary are at any time subject to special care and to interprocessor communication in order to properly compute the flux at these faces.

5 Application The system of vortices in the wake of airplanes continues to exist for a long period of time and this strongly influences the takeoff and landing frequency on an airport. It is possible to detect wake vortices as far as 100 wing spans behind the airplane, which is a hazard to following airplanes. In the framework of the collaborative research center SFB 401, one goal was to induce instabilities into the system of vortices to accelerate their collapse. The effects of different measures taken in order to destabilize the vortices have been examined in a water tunnel. A model of a wing was mounted in a water tunnel and the velocity components in the area behind the wing were measured using particle image velocimetry. It was possible to conduct measurements over a length of 4 wing spans. The experimental analysis of a sys-


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Fig. 5 Parallel simulation of the Lamb-Oseen vortex on 16 processors. Slices of the computational grid after 5,466 time steps (corresponding to a computed real time of 0.27 s)

tem of vortices far behind the wing poses great difficulties due to the size of the measuring system. Numerical simulations are not subject to such severe constraints and therefore Quadflow is used to examine the behavior of vortices far behind the wing. To minimize the computational effort, the grid adaptation adjusts the refinement of the grid with the goal to resolve all important flow phenomena, while using as few cells as possible. Figure 5 shows the simulation of a Lamb-Oseen vortex. More details on the simulation can be found in [3].

References 1. Ballmann, J.: Flow modulation and fluid-structure-interaction at airplane wings. Numerical Fluid Mechanics and Multidisciplinary Design, vol. 84, Springer, Berlin (2003) 2. Bramkamp, F., Lamby, Ph., Müller, S.: An adaptive multiscale finite volume solver for unsteady and steady state flow computations. J. Comp. Phys. 197(2), 460–490 (2004) 3. Brix, K., Melian, S., Müller, S., Schieffer, G.: Parallelisation of multiscale-based grid adaptation using space-filling curves. ESAIM Proc. 29, 108–129 (2009) 4. Müller, S.: Adaptive Multiscale Schemes for Conservation Laws. Volume 27 of Lecture Notes on Computational Science and Engineering, Springer, Berlin (2003) 5. Müller, S.: Multiresolution schemes for conservation laws. In: DeVore, R., Kunoth, A. (eds.) Multiscale, Nonlinear and Adaptive Approximation. Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday, pp. 379–408. Springer, Berlin (2009) 6. Sagan, H.: Space-Filling Curves. Springer, Berlin (1994) 7. Schröder, W.: Summary of Flow Modulation and Fluid-Structure Interaction Findings-Results of the Collaborative Research Center SFB 401 at the RWTH Aachen University, Aachen, Germany, 1997–2008. Volume 109 of Numerical Fluid Mechanics and Multidisciplinary Design, Springer, Berlin (2010) 8. Zumbusch, G.: Parallel multilevel methods. Adaptive Mesh Refinement and Load balancing. Advances in Numerical Mathematics, Teubner, Wiesbaden (2003)

Part XXII

Immersed Boundary Method

Recent Advances in the Development of an Immersed Boundary Method for Industrial Applications M.D. de Tullio, P. De Palma, M. Napolitano, and G. Pascazio

Abstract This paper provides some recent developments of an immersed boundary method for solving flows of industrial interest at arbitrary Mach numbers. The method is based on the solution of the preconditioned compressible Favreaveraged Navier – Stokes equations closed by the k-ω low Reynolds number turbulence model. A flexible local grid refinement technique is implemented on parallel machines using a domain-decomposition approach and an edge-based data structure. Thanks to the efficient grid generation process, based on the ray-tracing technique, and the use of the METIS software, it is possible to obtain the partitioned grids to be assigned to each processor with a minimal effort by the user. This allows one to by-pass the very time consuming generation process of a body-fitted grid.

1 Introduction The immersed boundary (IB) method is emerging as a very appealing approach for solving flows past very complex geometries, like those occurring in most industrial applications. Its main, very significant, feature is the use of a Cartesian grid embodying the complex boundaries of the flow domain, which allows one to generate the computational mesh within a few minutes, whereas a very complicated body fitted grid may require several hours or even days of manpower. The IB technique was originally developed for incompressible flows, see, e.g. [5] and the references therein, using non-uniform Cartesian grids to take advantage of simple numerical algorithms. More recently, the authors have contributed to the extension of the IB method to the preconditioned compressible Navier – Stokes (NS) equations in order to solve complex flows at any value of the Mach number [7], and equipped it with a local mesh refinement procedure to resolve boundary layers and regions with high flow gradients (e.g., shocks) [9]. In this work, some recent improvements and extensions of the method of [9] are presented, together with some new interesting results. M.D. de Tullio (B) CEMeC & DIMeG, Politecnico di Bari, 70125 Bari, Italy e-mail: [email protected]


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2 Numerical Method The basic numerical method uses a dual-time-stepping approach to solve unsteady flows. At every physical time level, the unsteady residual is reduced to any desired level by iterating in pseudo-time. When a steady solution is sought, the physical time derivative is removed and the steady residual is reduced by iterating in pseudotime [9]. Here only the new features are briefly described. Firstly, the use of an edge-based data structure allows one to refine the grid locally in any desired direction; this is a great advantage with respect to the previous approach [9], in which the grid had to be refined direction-by-direction throughout the entire flowfield, so that a successive coarsening step had to be applied far away form the high-gradient regions. Moreover, an anisotropic local grid refinement is performed, namely, each cell can be refined independently in each Cartesian direction. This feature complicates the grid topology but renders the approach more flexible to handle complex geometries with a remarkable reduction of the memory requirement with respect to a standard OCTREE data structure. In more details, starting from a grid with uniform mesh size, a locally refined grid is generated by recursively halving the mesh size at the immersed boundary region, until an assigned target value is reached [3]. This automatic refinement is based on the following strategy. A tag function, generated using the ray tracing technique, is used to mark the cells inside and outside the immersed body: an integer value ±1 is assigned to “fluid” and “solid” cells, respectively. The gradient of this function is different from zero only at the immersed boundary and depends on the local grid size. The components of this gradient in the three Cartesian directions are used to select the cells to be refined. The grid is refined until a user specified resolution is achieved at the boundary. In addition, one can refine other regions of the computational domain away from the immersed boundary, choosing the local resolution of the mesh, such as wake or bow-shock regions. Secondly, the method has been equipped with Sutherland’s law and temperaturevariable gas properties to handle hypersonic flows. Thirdly, a heat conduction solver has been combined with the flow one so as to solve conjugate-heat-transfer problems, see [10], for details. Finally, the code is parallelized implementing the communication exchange among the processors based on the MPI protocol. Employing the software METIS [4], the mesh is divided into a number of blocks defined by the user, balancing the number of computational cells among them. Following a domain decomposition approach, each block is assigned to a CPU which performs the integration of the NS equations in parallel, exchanging the needed information with its neighboring processors. All of the grid properties (coordinates, metrics, pointers for the communications among cells and among processors, etc.) are allocated according to the edge-based data structure and the data are provided in output files to be read by each processor. All operations are performed automatically and the user needs only to establish the number of processors employed for the computation.

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3 Results All of the results presented in this paper aim at validating the new features of the method, with particular attention to the domain decomposition and the code parallelization. The transonic flow past the unmanned space vehicle FTB-1 tested by the Italian Center for Aerospace Research (CIRA) has been computed at first [8]. The flight configuration corresponds to the launch experiment in which the vehicle falls from an altitude of 20 km; 30.8 s after the launch, the Mach number is equal to 0.94 and the incidence angle is 7.24 (zero elevon and rudder deflections have been assumed). The computational grid containing about 3.5 million cells is locally refined in the leading edge region of the wings and around the nose of the vehicle as shown by Fig. 1 which provides the local views of the grid at the midplane (y = 0) and at the plane y = 1 m. The grid has been generated automatically and partitioned into sixteen balanced blocks in about 15 min on a single core of an Intel Xeon X5560 @ 2.80 GHz processor The steady version of the code provided a residual drop of two orders of magnitude within about six CPU hours. Obviously, the residual cannot be reduced further, insofar as the flow has several unsteady features which need to be time resolved. Nevertheless, the computed complex shock system near the wings and in the rear part of the fuselage has been captured well with respect to the experimental data provided by CIRA [8]. Figure 2 provides the speed contours at same two longitudinal planes. The position and strength of the shocks are in good agreement with the data visualized in a wind tunnel at CIRA [8]. This test case demonstrates the flexibility of the proposed parallel IB numerical strategy and its applicability to very complex flow fields.

Fig. 1 USV flow: grid at y = 0 (left) and y = 1 m (right)

Then, the classical hypersonic flow over a 2D compression ramp [6] has been computed. The ramp geometry is employed to study the effects of flap deflection on the flow past a space vehicle. This configuration shows the typical features of shock wave-boundary layer interaction with flow separation and re-attachment. Slip velocity conditions are computed using the first-order relation provided in [1]. The distance xc , between the leading edge of the flat plate and the ramp corner

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Fig. 2 USV flow: velocity magnitude contours at y = 0 (left) and y = 1 m (right)

is equal to 71.4 mm and the ramp angle and length are 35◦ and 71.4 mm, respectively [6]. The flow of nitrogen gas is considered with the following free-stream conditions [6]: ρ∞ = 1.401 × 10−4 kg m−3 , V∞ = 1521 m s−1 , T∞ = 9.06 K . The corresponding free-stream Mach number and Reynolds number are 24.8 and 12,020, respectively. The wall is perfectly diffusing with the wall temperature set at 403.2 K . Two-dimensional computations have been performed using a locally refined grid with about 80,000 cells clustered at the leading edge of the plate and in the recirculation region. The height of the first cell along the wall is 0.02 mm. About 20 wall-clock CPU minutes are needed to generate the grid and to drop the steady residual by two orders of magnitude using two 8-core Intel Xeon @2.80Ghz processors. Figure 3 reports the streamlines (superposed to the density contours) and the pressure coefficient distribution which is in good agreement with the Direct Monte-Carlo Simulation of Moss [6]. 1.6 1.4

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Finally, the code has been applied to the the simulation of a highly-loaded cooled two-dimensional turbine cascade. The geometry of the blade, known as the T106 turbine cascade [2], has been modified by adding three cooling channels. The flow is subsonic, with isentropic exit Mach number equal to 0.3, inlet flow angle equal to

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37.7, and Reynolds number, based on the chord length and on the exit conditions, equal to 3 × 105 . Air and stainless steel are considered for the fluid and for the solid, respectively. At the inlet boundary points, the total pressure and temperature are assigned, together with the flow direction, whereas only the static pressure is prescribed at the outlet points. The two side holes have assigned wall temperature, equal to Tc = 200 K , whereas, cooling air flows from the central hole into a secondary channel so as to form a cooling film on the suction side of the blade. The inlet flow condition for the secondary air are set at midspan, where total temperature and pressure conditions are imposed corresponding to an inlet temperature Tc = 200 K and an inlet velocity normal to the endwall, vc = 5 m s−1 . Thanks to the versatility of the present IB approach, the complete geometry of the blade can be discretized easily and efficiently. The computational grid, using about 66,000 cells (33,700 in the solid region), shown in Fig. 4(a), is refined at the leading edge of the blade, at the region of maximum curvature, and near the cooling holes, see Fig. 4b. Figure 5a, b provide the computed temperature countours in the solid and in the fluid, and the velocity-vector field in and around the main central cooling channel. This test case demonstrates the capability of the present method to solve conjugate-heat-transfer problems of industrial interest. 300 295 290 285 280 275 270 265 260 255 250 245 240 235 230 225 220 215 210 205 200

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4 Conclusions and Future Work This work has presented some recent improvements and applications of an immersed-boundary method suitable for solving industrial problems from incompressible to hypersonic flow conditions. The method has been tested versus three applications of aerospace and industrial interest. Current work aims at coupling the unsteady version of the proposed methodology with an integral method for solving the Ffowcs Williams and Hawkings equations, so as to predict aerodynamic noise. Acknowledgements This work has been supported by the MIUR and the Politecnico di Bari, Cofinlab 2000 and by PRIN-2007 grants.

References 1. Hadjiconstantinou, N.G.: The Limits of Navier–Stokes Theory and Kinetic Extensions for Describing Small Scale Gaseous Hydrodynamics. Phys. Fluids, 18, 111301 (2006) 2. Hoheisel, H., Kiock, R., Lichtfuss, H.J., Fottner, L.: Influence of free-stream turbulence and blade pressure gradient on boundary layer and loss behaviour of turbine cascades. ASME J. Turbomach. 109, 210219 (1987). 3. Kang, S., Iaccarino, G., Ham, F.: DNS of buoyancy-dominated turbulent flows on a bluff body using the immersed boundary method. J. Comput. Phys. 228, 3189 (2009) 4. Karypis, G., Kumar, V.: A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comput. 20, 359 (1998) 5. Mittal, R., Iaccarino, G.: Immersed Boundary Methods. Annu. Rev. Fluid Mech. 37, 239 (2005) 6. Moss, J.N., Rault, D.F.G., Price, J.M.: Rarefied gas dynamics: Space science and engineering. In: Shizgal, B.D., Weaver, D.P. (eds.) 160 of Progress in Aeronautics and Aeronautics, AIAA, Washington, pp. 209–220, 1993 7. De Palma, P., de Tullio, M.D., Pascazio, G., Napolitano, M.: An immersed boundary method for compressible viscous flows. Comput. Fluids 35, 693 (2006) 8. Rufolo, G.C., Marini, M., Roncioni, P., Borrelli, S.: In flight aerodynamic experiment for the unmanned space vehicle FTB-1, First CEAS European Air and Space Conference, Berlin, 10–13 September 2007 9. de Tullio, M., De Palma, P., Iaccarino, G., Pascazio, G., Napolitano, M.: An immersed boundary method for compressible flows using local grid refinement. J. Comput. Phys. 225, 2098– 2117 (2007) 10. de Tullio, M.D., Latorre, S.S., De Palma, P., Napolitano, M., Pascazio, G.: An Immersed Boundary Method for Conjugate Heat Transfer Problems. ASME 2010 FEDSM2010ICNMM2010, Montreal, Canada, 2–4 August 2010

An Overview of the LS-STAG Immersed Boundary Method for Viscous Incompressible Flows Olivier Botella and Yoann Cheny

Abstract The LS-STAG method is an immersed boundary method for viscous incompressible flows based on the staggered MAC arrangement for Cartesian grids, where the irregular boundary is sharply represented by its level-set function. The level-set function enables us to compute efficiently all relevant geometry parameters of the so-called “cut-cells”, i.e. the cells that are cut by the immersed boundary, reducing thus the bookkeeping associated to the handling of complex geometries. One of the main features of the LS-STAG method is the use of a consistent and unified discretization of the flow equations in both Cartesian and cut-cells, which has been obtained by enforcing the strict conservation of global invariants of the flow such as total mass, momentum and kinetic energy in the whole fluid domain. After a short discussion on the salient features of the LS-STAG method, we will present a recent application: the computation of viscoelastic flows in planar contraction geometries.

1 Introduction This paper presents an overview of the LS-STAG method [2, 3], which is a novel immersed boundary (IB) method for flows in moving irregular geometries on fixed Cartesian grids. In IB methods (see [5] for a recent review), the irregular boundary is not aligned with the computational grid, and the treatment of the cut-cells, cells of irregular shape which are formed by the intersection of the Cartesian cells by the immersed boundary, remains an important issue. Indeed, the discretization in these cut-cells should be designed such that: (a) the global stability and accuracy of the original Cartesian method are not severely diminished and (b) the high computational efficiency of the structured solver is preserved. Two major classes of IB methods can be distinguished on the basis of their treatment of cut-cells. Classical IB methods such as the momentum forcing method [5], use a finite volume/difference structured solver in Cartesian cells away from the O. Botella (B) LEMTA, Nancy-University, CNRS, 54504 Vandœuvre, France e-mail: [email protected]


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irregular boundary, and discard the discretization of flow equations in the cutcells. Instead, special interpolations are used for setting the value of the dependent variables in the latter cells. Thus, strict conservation of quantities such as mass, momentum or kinetic energy is not observed near the irregular boundary. The most severe manifestations of these shortcomings is the occurrence of nondivergence free velocities or unphysical oscillations of the pressure in the vicinity of the immersed boundary. Numerous revisions of these interpolations are still proposed for improving the accuracy and consistency of this class of IB methods. A second class of IB methods (also called cut-cell methods or simply Cartesian grid methods [5]), aims for actually discretizing the flow equations in cut-cells. The discretization in the cut-cells is usually performed by ad hoc treatments which have more in common with the techniques used on curvilinear or unstructured bodyconformal grids than Cartesian techniques. Such treatments of the cut-cells generate a non-negligible bookkeeping to discretize the flow equations and actually solve them, and it is difficult to evaluate the impact of these treatments on the computational cost of the flow simulations. The purpose of this communication is to present an overview of the LS-STAG method [2, 3] for incompressible viscous flows which takes the best aspects of both classes of IB methods. This IB method, which is based on the symmetry preserving finite-volume method of Verstappen and Veldman [8] for non-uniform Cartesian grids, has the ability to preserve up to the cut-cells the conservation properties (for total mass, momentum and kinetic energy) of the original MAC method. After a short discussion on the salient features of the method, we will present its most recent application: the computation of viscoelastic flows in planar contraction geometries governed by the Oldroyd-B constitutive equation [7].

2 Basics of the LS-STAG Method for Newtonian and Viscoelastic Flows in Irregular Geometries The LS-STAG method presented in [2, 3] is a finite volume method for computing fluid flows in the irregular fluid domain Ω f = Ω \ Ω ib , where Ω ib is a solid domain immersed in the rectangular computational domain Ω (see Fig. 1). As shown in this figure, the irregular boundary Γ ib = ∂Ω ib is implicitly represented by its signed distance function φ(x, y) (i.e. the level-set function [6]), which is discretized at the vertices of the rectangular cells. The level-set function enables us to compute efficiently all relevant geometry parameters of the cut-cells (such as their volume, projected areas, boundary conditions, . . . ), reducing thus the bookkeeping associated to the handling of complex geometries. Figure 2 shows the 3 generic types of cut-cells which are present in the LSSTAG mesh. For building our discretization in each type of cut-cells, we have required that 5 quantities of the flow such as 5 our scheme strictly conserves global total mass Ω f ∇ · vdV , total momentum P(t) = ρ Ω f v dV and total kinetic energy 5 Ec (t) = 12 Ω f |v|2 dV (when viscosity η becomes negligible), which are crucial

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Fig. 1 Staggered arrangement of the variables near the trapezoidal cut-cell Ωi, j on the LS-STAG mesh. The control volume for u i, j is shown with dashed lines, and non-homogeneous velocity boundary conditions are discretized at the vertices () of the cut-cells

(a) Northeast Pentagonal Cell

(b) North Trapezoidal Cell

(c) Northwest Triangle Cell

Fig. 2 Sketch of the 3 generic cut-cells and location of the normal and shear stresses

properties for obtaining physically realistic numerical solutions, see [8] and references therein. It is well known that the original MAC method on uniform Cartesian meshes with central differencing conserve these global invariants. On the other hand, for more general grid systems or higher-order methods the construction of “globally conservative” methods is not a trivial task, and one needs to enforce the conservation properties to the discretization scheme [8]. For example, let us consider the following semi-discretization of the incompressible Navier-Stokes equations:

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ρ

d (M U ) + C [U ]U + G P − ηK U = 0, dt DU = 0,

(1a) (1b)

where the diagonal mass matrix M is built from the volume of the fluid cells, matrix C [U ] represents the discretization of the convective fluxes, G is the discrete pressure gradient, K represents the Newtonian stresses, and D is the discrete divergence. For simplicity, we have discarded here the influence of the boundary conditions, but the complete discussion can be found in Ref. [3]. The budget for the discrete kinetic energy Ech (t) is obtained from the discrete momentum equation as: C [U ]T + C [U ] η(K T + K ) dEch = −U T U − P TG TU − U T U, dt 2 2

(2)

and if we require that this discrete budget mimics the dissipation of energy by the viscous forces in the whole fluid domain, then we have to require that the discretization of the convective terms leads to a skew-symmetric matrix: C [U ] = −C [U ]T , the pressure gradient has to be dual to the divergence operator: G = −D T , and the viscous matrix K T + K should be positive definite. In Ref. [3], we have developed the LS-STAG immersed boundary method such as the convective, pressure and viscous fluxes in the cut cells of Fig. 2 are unambiguously determined by these requirements, and such that non-homogeneous boundary conditions at the immersed boundary are naturally embedded. On Cartesian grids, the LS-STAG method reduces to the Cartesian method of Verstappen and Veldman [8]. From the algorithmic point of view, one of the main consequences is that the LS-STAG discretization preserves the 5-point structure (in 2D) of Cartesian methods, and allows the use of efficient black-box solvers for structured grids, where no ad hoc modifications had to be undertaken for taking account of the immersed boundary. Recently, the LS-STAG method has been extended in [2] to the computation of viscoelastic flows [7]. The shear stress tensor is decomposed in a Newtonian (solvent) elastic part as τ = ηs ∇v + τe , where the extra-stress tensor x x x yand τe τe τe = x y x y is governed by the Oldroyd-B transport equation, whose generic τe τe expression may be written as: λ

d dt

Ω

τ dV +

(v · n) τ dS =

Ω

Sτ dV,

(3)

5 where λ is the elastic characteristic time and Ω Sτ dV is a volumic terms coming from the definition of the upper-convected derivative. The nondimensional number that measures the level of elasticity of the flow is the Weissenberg number We = λU/L, where L and U are characteristic length and velocity. The solution of the viscoelastic flow system (1) and (3) presents several numerical challenges [7], one of the most severe being related to the velocity-pressurestress coupling. Indeed, as for the velocity and pressure coupling for Newtonian

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flows, the discretization of the velocity and stress has to be compatible for preventing nonphysical node-to-node oscillations of the extra-stress variables. For a finitevolume method on a Cartesian grid, a compatible discretization is usually achieved by an adequate staggering of the normal and shear stress unknowns at the location of their Newtonian counterparts. For enforcing this property in the cut-cells of the xy LS-STAG mesh, we have to position the viscoelastic shear-stress unknowns τi, j at the vertices of the cut-cells, and the normal stresses at their centroid, see Fig. 2. All details of the discretization of the Oldroyd-B system (3) are given in [2]. Firstly, staggered control volumes are defined for the normal and shear stresses in the cut-cells and, for the volumic integrals in (3) which were absent from the Navier-Stokes equations, novel quadratures with good conservation properties are constructed for the 3 generic cut-cells of Fig. 2.

3 Numerical Results for Viscoelastic Oldroyd-B Flows In Ref. [3], the accuracy and the robustness of the LS-STAG method has been assessed on benchmark Newtonian flows at low to moderate Reynolds number (Couette Taylor flow, flows around cylinders, . . . ), including the case where the complex geometry is moving. This last case is one of the most appealing features of IB methods, since the computations are performed on fixed grids without domain remeshing at each time-step. In the following, we present some unpublished results from [2] that concern a popular benchmark for viscoelastic fluids: the creeping flow of an Oldroyd-B fluid in a 4:1 planar contraction with rounded re-entrant corner [1, 4], see the sketch in Fig. 3 (left). Numerical results up to We = 8 have been obtained on the 150 × 71 mesh shown in Fig. 3 (right), which has 64% of fluid cells. Figure 4 shows that the LS-STAG method predicts accurately the reduction of the salient vortex when the level of elasticity increases, and that the contours are free from any spurious oscillations thanks to the fully staggered arrangement of the flow variables. Table 1 reports quantitative results of the flow for increasing values of the elasticity level. These results are compared to those obtained with conformal grid methods [1, 4], and where they are available, a good agreement is met with the literature.

Fig. 3 Sketch of the 4:1 planar contraction flow and close up of the mesh near the re-entrant corner

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Table 1 Properties of the corner vortex and Couette coefficient C for increasing values of We We=2

We=4

We=6 We=8

LS-STAG Ref. [1] Ref. [4] LS-STAG Ref. [1] Ref. [4] LS-STAG Ref. [4] LS-STAG

XR

Ψ × 10−3

C

1.095 − − 0.923 − − 0.764 − 0.602

0.368 0.342 0.48 0.249 0.225 0.33 0.207 0.26 0.236

−1.469 −1.72 −0.69 −2.843 −3.54 −1.44 −3.154 −2.27 −2.834

References 1. Aboubacar, M., Matallah, H., Webster, M.F.: Highly elastic solutions for Oldroyd-B and PhanThien/Tanner fluids with a finite volume/element method: Planar contraction flows. J. NonNewtonian Fluid Mech., 103, 65–103 (2002) 2. Cheny, Y.: La Méthode LS-STAG: une nouvelle Approche de type Frontière Immergée/LevelSet pour la Simulation d’Écoulements Visqueux Incompressibles en Géométries Complexes. Application aux Fluides Newtoniens et Viscoélastiques. PhD Thesis, Nancy-University (2009) 3. Cheny, Y., Botella, O.: The LS-STAG method: A new immersed boundary / level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys., 229, 1043–1076 (2010) 4. Matallah, H., Townsend, P., Webster, M.F.: Recovery and stress-splitting schemes for viscoelastic flows. J. Non-Newtonian Fluid Mech. 75, 139–166 (1998) 5. Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005) 6. Osher, S., Fedkiw, R.P.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New-York (2003) 7. Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002). 8. Verstappen, R.W.C.P. Veldman, A.E.P.: Symmetry-preserving discretization of turbulent flow. J. Comput. Phys., 187, 343–368 (2003)

A Two-Dimensional Embedded-Boundary Method for Convection Problems with Moving Boundaries Yunus Hassen and Barry Koren

Abstract A 2D embedded-boundary algorithm for convection problems is presented. A moving body of arbitrary boundary shape is immersed in a Cartesian finite-volume grid, which is fixed in space. The boundary surface is reconstructed in such a way that only certain fluxes in the immediate neighbourhood indirectly accommodate effects of the boundary conditions valid on the moving body. Over the majority of the domain, where these boundary conditions have “no” effect, the fluxes are computed using standard schemes. Examples are given to validate the method.

1 Introduction Recently, immersed-boundary methods have been favourably popularised by their relatively simple ideas and ease of implementation. The immersed-boundary method, also synonymously known as embedded-boundary method, in general, is a method in which boundary conditions are indirectly incorporated into the governing equations. It is very suitable for simulating flows around flexible, moving and/or complex bodies (see [5] for a comprehensive review). In this work, we present a new embedded-boundary approach for advection problems. As is standard in the immersed-boundary methods, moving bodies are embedded in a fixed, Cartesian grid. We employ the method of lines: a higher-order, cell-averaged, fixed-grid, finite-volume method for the spatial discretization, and the explicit Euler scheme for the time integration. The essence of the present method is that body geometries are, without loss of generality, effectively simplified and their presence is restricted to a minimal zone in the computational region so that standard discretization schemes can be readily applied elsewhere. The boundary conditions valid on a possibly moving body are indirectly accommodated by specific fluxes in the vicinity of the boundary. Y. Hassen (B) Centrum Wiskunde & Informatica, Amsterdam, The Netherlands; Faculty of Aerospace Engineering, TU Delft, The Netherlands e-mail: [email protected]


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2 The Embedded Boundary Method As in the previous one-dimensional work [1], our approach uses a finite-volume discretization that embeds the boundary of a body in a regular, fixed grid. Dividing the current 2D computational domain D, of dimension *x × * y , into N x × N y uniform, rectangular finite volumes that are fixed in space, we have: 0 is equally divided into Nt time steps of size τ = T /Nt . Having generated the (Cartesian) grid, Eq. (1), the body is immersed inside the grid. To obtain discrete embedded boundaries (EBs), at a given time t n , n = 0, 1, · · · , Nt , firstly, the finite volumes that contain (a part of) the boundary of the immersed body are identified, and then the points of intersection of the boundary of the immersed body with the faces of these computational cells, xnB , are

detected. That is, the coordinates of the boundary points xk,n := x Bk,n , yBk,n , B k = 1, 2, · · · , NB , where NB is the total number of boundary-face intersections, are computed. Care is required in making the underlying uniform fixed grid detect boundary points that lie exactly at or very close to grid vertices. Grid vertices are shared by more than one control volume, and a boundary point lying exactly at a grid vertex, or in the immediate neighbourhood, is, due to the round-off error, arbitrarily assigned to any of the cells that share the vertex. The prevailing arbitrariness can lead to erroneous absence of an EB in a cell, due to discount of a second boundary point (or two boundary points altogether) within the cell. This undesirable scenario can be remedied by taking the precision of the machine into account; see [2] for details.

2.1 Determination of EBs Once the boundary points have been detected, the actual boundary of the body is readily degenerated as a piece-wise continuous, closed or open, poly-line, with NB and NB − 1 segments, respectively. This representation facilitates explicit association of each side/segment of the polygon/poly-line with individual control volumes, resulting in one discrete EB, at most, in a cell. The manner in which a generic 2D EB segment, situated in a cell (Fig. 1a), is projected onto the grid (coordinate) directions is crucial. To take advantage of the 1D method proposed in [1], we resort to dimensional splitting. The procedure is illustrated in Fig. 1 and described in detail in [2].

A Two-Dimensional Embedded-Boundary Method

δy

615

δy δx

δx

(a)

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Fig. 1 Alignment of a two-dimensional EB, situated in a cell, with the relevant grid (coordinate) direction. (a) Continuous/discrete EB. (b) δx ≤ δy. (c) δx > δy

To project the discrete EB, shown in Fig. 1a, into the relevant grid direction and to get a single orthogonal EB in a cell, two geometrical properties – orientation and location – are used. The oblique discrete EB (Fig. 1a) is next rotated parallel to the grid direction to which it is closest (Fig. 1b or Fig. 1c). The actual

location of the orthogonalized EB, at t n , inside a cell is represented by β n = βxn , β yn , a normalised variable that discerns the orthogonalized EB’s position relative to the left- or bottom-face of the cell, and it is determined by the area, subset in the cell by the non-orthogonalized EB. Having all the discrete EBs in the domain aligned with the relevant grid direction and appropriately positioned inside a cell, we achieve the desired sub-cell resolution of the immersed body boundary. In the latter respect, the present method essentially differs from the stair-case approach wherein the boundary is projected on cell faces [5].

2.2 Merging of EBs Occasionally, it might arise a scenario with two successive cells, along a column or row of cells, each having orthogonalized EBs in the same direction. Technically, with the outlined procedure, this is a natural outcome and it can be accommodated. However, this situation may be non-physical. It might cause “numerical spray” and, as such, it perturbs an evolving solution, not to mention the algorithmic (flux-computation and time-stepping) complications it creates during implementation. In the event of two such distinct, orthogonalized EBs, we can effectively get rid of the “numerical spray” by properly merging these EBs, obtaining a single equivalent EB in one of the cells. The “numerical spray” is superseded by reuniting it with its “parent material” that originally gave it off. Therefore, the merging is done in the direction of the “parent material” and such that the conservation law is satisfied. Further details are given in [2].

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3 Finite-Volume Discretization A multi-dimensional convection problem, in conservation form, with the associated initial condition, can be written as: ∂c + ∇ · F = 0, in D ∈ R2 × (0, T ], ∂t c(x, t = 0) = c0 (x), in D ∈ R2 ,

(2a) (2b)

where c(x, t) is the scalar field that is convected by the flow field u = (u(x) , v(x))T . The components of the flux vector F = ( f (u, c), g(v, c))T are defined, at t n , as: f n (u, c) := u(x) c(x, t n ) and g n (v, c) := v(x) c(x, t n ). For a cell-averaged discrete solution in Di, j , at t n , i.e., ci,n j , an integral form of (2a) can be formulated. Assuming fluxes to be constant along cell faces, we have the semi-discrete equation: dci, j + hx h y dt

f

1 (t) − f 1 (t) i+ 2 , j i− 2 , j

+ g

1 (t) − g 1 (t) i, j+ 2 i, j− 2

= 0. (2c)

Certain fluxes in the immediate neighborhood of an orthogonalized EB are modified to accommodate the corresponding embedded-boundary conditions, see [1] for details. Elsewhere, where the EBs have “no” effect, the standard MUSCL scheme [3, 4] is used. With a very small Courant number ν, Eq. (2c) yields a temporally accurate solution, using the forward Euler scheme: ci,n+1 j

=

ci,n j

τ − hx h y

fn 1 i+ 2 , j

−

fn 1 i− 2 , j

τ − hx h y

gn 1 i, j+ 2

−

gn 1 i, j− 2

.

(2d)

4 Numerical Examples To validate the algorithm presented in this article, we consider two test cases, a translating rectilinear discontinuity and a revolving cylindrical discontinuity, with prescribed 2D flow fields inside a rectangular domain. Settings of the problems are depicted in Fig. 2. Here, we take *x = * y = 2, with D = [−1, 1] × [−1, 1].

4.1 A Translating Rectilinear Discontinuity Consider a rectilinear discontinuity of arbitrary orientation ϑ ∈ [0, π/2], initially situated at the bottom-left corner of the domain, and moving in a uniform 2D flow field of velocity u = (cos ϑ, sin ϑ)T . The flow field, shown in Fig. 2 (left), is normal to the discontinuity. The discontinuity, which goes with the flow, is assumed to model a rigid, infinitely thin plate that separates two quantities of different values, i.e., c = 1


617 y

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EB

v

ω

x

0< ϑ < π 2

x

ϑ

u

ϑ u

v

Fig. 2 Domain, flow fields and problem settings. A rectilinear (left) and cylindrical (right) discontinuties

(at the upstream side) and c = 0 (at the downstream side). These solution values are taken as embedded boundary conditions, i.e., cEBl = 1 and cEBr = 0, to be used in the relevant fixed-grid fluxes in the immediate neighbourhood of the embedded boundary. Figure 3 shows results for the translating rectilinear discontinuity, of arbitrary √ orientation ϑ = π/6, on a grid (N x , N y ) = (20, 20) and at final time T = 2. The results obtained with the current method appear to be significantly more accurate than those obtained with the standard method. Results for other orientations in the range ϑ ∈ [0, π/2] are given in [2]. 1

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4.2 A Revolving Cylindrical Discontinuity Consider a cylindrical discontinuity of radius R = 0.2, and unit height, initially located at (x, y) = (1/2, 1/2), shown in Fig. 2 (right), which revolves with a circular flow-field of velocity u = (−ωy, ωx)T , where ω = 2π is the angular velocity (a solid-body rotation).

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Similarly, as in Sect. 4.1, the discontinuity, which goes with the flow, is assumed to model a rigid, infinitely thin-walled cylinder that separates two quantities of different values, i.e., c = 1 and c = 0, inside and outside the cylinder, respectively. The solution values c = 1 and c = 0 are appended to the boundary of the immersed cylinder; cEBl = 1 at the inner, and cEBr = 0 at the outer side of the cylinder’s wall. Figure 4 shows the results for the revolving cylindrical discontinuity obtained on a grid of (N x , N y ) = (40, 40), at a final time T = 1 (i.e., after one full revolution). Clearly, the results of the current EB method, on this grid, have higher resolution than those of the standard method, but they are not yet monotone. 1 0.7

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5 Conclusion In this work, a new immersed-boundary method, which effectively embeds boundary conditions, valid on a moving body, only in certain fluxes in the immediate neighbourhood, has been introduced. The algorithm has been tested with two problems and the results obtained are promising. They have higher resolution compared to those obtained by standard methods, but are not yet entirely monotone. It is a simple and elegant algorithm, and we anticipate to use it for 2D Euler flows, which we foresee to consider next. Acknowledgements The first author’s research is funded by the Delft Centre for Computational Science & Engineering (DCSE), TU Delft.

References 1. Hassen, Y., Koren, B.: Finite-volume discretizations and immersed boundaries. In: Koren, B., Vuik, C. (eds.) Advanced Computational Methods in Science and Engineering, Lecturer Notes in Computational Science and Engineering, vol. 71, pp. 229–268. Springer, Heidelberg (2010) 2. Hassen, Y., Koren, B.: A two-dimensional embedded-boundary method for convection problems with moving boundaries. Report MAC-1003, CWI, Amsterdam (2010)


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3. Koren, B.: A robust upwind finite-volume method for advection, diffusion and source terms. In: Vreugdenhil, C.B., Koren, B. (eds.) Numerical Methods for Advection-Diffusion Problems, Notes on Numerical Fluid Mechanics, vol. 45, pp. 117–138. Vieweg, Braunschweig (1993) 4. van Leer, B.: Upwind-difference methods for aerodynamic problems governed by the Euler equations. In: Engquist, B.E., Osher, S., Somerville, R.C.J. (eds.) Large-Scale Computations in Fluid Mechanics, Lectures in Applied Maths, vol. 22.2, pp. 327–336. Am. Math. Soc., Providence, RI (1985) 5. Mittal, R., Iaccarino, G.: Immersed boundary methods. Ann. Rev. Fl. Mech. 37, 239–261 (2005)

A Second-Order Immersed Boundary Method for the Numerical Simulation of Two-Dimensional Incompressible Viscous Flows Past Obstacles François Bouchon, Thierry Dubois, and Nicolas James

Abstract We present a new cut-cell method, based on the MAC scheme on Cartesian grids, for the numerical simulation of two-dimensional incompressible flows past obstacles. The discretization of the nonlinear terms, written in conservative form, is formulated in the context of finite volume methods. While first order approximations are used in cut-cells the scheme is globally second-order accurate. The linear systems are solved by a direct method based on the capacitance matrix method. Accuracy and efficiency of the method are supported by numerical simulations of 2D flows past a cylinder at Reynolds numbers up to 9,500.

1 Introduction Avoiding the use of curvilinear or unstructured body-conformal grids, immersed boundary (IB) methods provide efficient solvers, in terms of computational costs, on Cartesian grids for flows in complex geometries. IB methods can be classified in two groups. Classical IB methods add in the momentum equation a forcing term accounting for the presence of an obstacle in the computational domain. Cut-cell methods discretize the momemtum and continuity equations in mesh cells cut by the solid. The scheme proposed in this paper lies in this class of IB methods and differs from other cut-cell methods in the treatment of the diffusive and convective terms in cut-cells. The scheme is globally second-order accurate for the velocity and pressure variables. This paper is organized as follows. The first section is devoted to the description of the problem. Then the IB/cut-cell method is detailed and finally some numerical simulations are given.

F. Bouchon (B) Laboratoire de Mathématiques, Université Blaise Pascal and CNRS (UMR 6620), Campus Universitaire des Cézeaux, 63177 Aubiere, France e-mail: [email protected]


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2 The Settings of the Problem Let Ω be a rectangular domain. We consider an irregular fluid domain Ω F which is embedded in the computational domain Ω and its complement Ω S , the solid domain. The interface between solid and fluid is denoted Γ . The decoupling between velocity and pressure variables is achieved by applying a second-order (BDF) projection scheme to the incompressible Navier-Stokes equations. First, the prediction step consists in computing the velocity field uk+1 which is solution of the following equation: 3u˜ k+1 − 4uk + uk−1 − u˜ k+1 /Re = −∇ pk + f k+1 − 2 div uk ⊗ uk 2δt + div uk−1 ⊗ uk−1

(1)

with appropriate boundary conditions on ∂Ω F . Then we solve the projection step: 2δt k+1 − pk , ∇ p 3 div uk+1 = 0, k+1 − u˜ k+1 |∂Ω .n = 0. u

uk+1 = u˜ k+1 −

(2) (3) (4)

3 The IB/Cut-Cell Method 3.1 Staggered Arrangement of the Unknowns As in Cheny and Botella [3], a signed distance to the obstacle d is used to represent solid boundaries in the computational domain. The pressure is placed at the center of every Cartesian cell either filled by the fluid or cut by the solid boundaries (see Fig. 1). The velocity components are placed at the middle of the part of the edges located in the fluid. Unlike for cells located in the fluid part of the computational domain, velocity and pressure are not aligned in cut-cells. Cell-face ratios ri,u j , ri,v j are calculated from the distance of the mesh point to the obstacle, denoted di, j . The interface Γh , linear in each cell, approaches the regular solid boundary Γ (see Fig. 1).

3.2 Discretization of the Prediction Step In fluid cells, the classical five-point stencil approximation of the viscous terms is used. In cells sharing a face with a cut-cell, we propose a first-order Finite Difference approximation. More precisely, we consider V = {O, N , S, E, W, P} with O the position of u i j , N , S, E, W are the location of unknowns close to O or on the

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vi j

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P

Fig. 2 Six points are used for the discretization of the diffusive term in mesh cells cut by the solid boundary

boundary, and P is arbitrarily chosen (see Fig. 2). Then, we search coefficients α M such that M∈V α M u(M) is a first-order approximation of u(O). This leads to a linear system of six equations with six unknowns. In fluid cells, a second-order centered approximation for the nonlinear terms is used. In cells sharing a face with a cut-cell, we propose a first-order Finite Volume approximation of nonlinear terms. The integral of the first component of the nonlinear term over a cut-cell K˜ i,u j = K i,u j ∩ ΩhF is expressed in terms of fluxes through cell edges, namely: K˜ i+ 1 , j 2

2 ∂x (u ) + ∂ y (uv) dx =

∂ K˜ i+ 1 , j

2 u n x + (uv)n y d S

(5)

2

E E N N B = Fi+1, j − Fi, j + Fi, j − Fi, j−1 + Fi, j . (6)

Second-order interpolations of velocity components are used to approximate the fluxes at the center of cut-edges (see Fig. 3). This leads to a pointwise first-order approximation of the convective terms.

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yj

ΩSh

FiNj FiBj FiEj

ui j

E Fi+1 j

yj−1 xi−1

xi

FiNj−1

xi+1

Fig. 3 Discretization of the convective term using fluxes reconstruction

3.3 Discretization of the Projection Step The continuity equation is decomposed as the net mass flux through each face of the computational cells. Due to the locations of velocity components, a secondorder approximation immediately. The discretization of (3) on a cut-cell is 0 follows supp u i, j + Di, j = 0, where the linear part of the discrete diver(Dobs u)i, j = Dobs gence is

0 u Dobs

i, j

u v v = h y ri,u j u i, j − ri−1, j u i−1, j + h x ri, j vi, j − ri, j−1 vi, j−1

(7)

supp

and the contribution to the divergence due to the boundaries is Di, j = *i, j g(M) . ni, j . Note that *i, j , M and ni, j are respectively the length, the middle point and the external normal of the edge of the cut-cell shared with the boundary (see Fig. 1). In the case of a fluid cell, this expression reduces to the standard MAC discretization. The velocity correction step requires the computation of the pressure gradients at the location of the velocity. For faces cut by solid boundaries, a second-order interpolation Pφ is used. As in the classical MAC scheme, a discrete Poisson-type equation for the pressure increment δp k+1 = p k+1 − pk is obtained by applying the discrete divergence operator to the velocity correction equations: 3 0 Pφ (Gδp k+1 ) = Dobs (u˜ k+1 ), Dobs 2 δt

(8)

with the classical discrete gradient (Gδpk+1 )i, j =

k+1 k+1 δpi+1, j − δp i, j

J

hx ,

k+1 J h . δpi,k+1 − δp y i, j j+1

(9)

It follows that the velocity correction uk+1 = u˜ k+1 −

2 δt Pφ (Gδp k+1 ) 3

(10)

A Second-Order Immersed Boundary Method

625

ensures that the incompressibility condition Dobs (uk+1 ) = 0 is satisfied in the whole domain up to computer accuracy.

3.4 Computational Efficiency The non-symmetric linear systems are efficiently solved by a direct method, based on the capacitance matrix method [2]. First, we solve a preprocessing step, requiring O(n 3 ) operations. Assuming that the obstacle does not move, this step is solved once per simulation. Then, at every time step, this technique allows to reduce the overall cost of the resolution to O(n 2 log(n)) operations, which is the number of operations needed to solve the linear systems corresponding to five-point stencil operators on the whole Cartesian mesh without obstacle. The method is tested on the Taylor-Couette flow between two concentric cylinders: second-order spatial convergence for velocity and pressure is found. In Fig. 4, we have reported the L ∞ error for the velocity, when the error is measured on the whole fluid computational domain. Unlike in [3], the second-order accuracy is also satisfied in cut-cells. Numerical simulations of 2D flows past a cylinder have been performed at Reynolds numbers up to 9,500. As it is shown on Fig. 5, an excellent agreement is found with the experimental results presented in Bouard and Coutenceau [1]. Streamlines of the flow past a cylinder at Re = 9,500 at time t = 0.75, 1.0 and 1.25 are represented on Fig. 5. We have also studied the flow past a NACA aerofoil at Re = 1,000. Like in [4], a Karman vortex street develops behind the obstacle (see Fig. 6): the flow is well resolved even near the sharp ending edge. For these numerical simulations, the mesh size near the obstacle is 1.6 10−3 . The value of the time step, satisfying a CFL stability condition, is 10−4 .

0,1

Error

0,01 0,001

0,0001 1e-05 0,01

0,1

h

Fig. 4 Error for the velocity versus grid size h. Present study (circles), Cheny and Botella results [3] (squares), first-order slope (dash) and second-order slope (dotted)

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Fig. 5 Evolution of the boundary layer: comparison with experimental results at Re = 9,500 (the upper figures are reproduced from [1] by permission of Cambridge University Press)

Fig. 6 Flow behind a NACA aerofoil at Re = 1,000

References 1. Bouard, R., Coutanceau, M.: The early stages of development of the wake behind an impulsively started cylinder for 40 < Re < 104 . J. Fluid. Mech. 101, 583–607 (1980) 2. Buzbee, B.L., Dorr, F.W., George, J.A., Golub, G.H.: The direct solution of the discrete Poisson equation on irregular regions. SIAM J. Num. Anal. 8, 722–736 (1971) 3. Cheny, Y., Botella, O.: The LS-STAG method: A new immersed boundary/level-set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties. J. Comput. Phys. 229–4, 1043–1076 (2010) 4. Daube, O., Loc, T.P., Monnet, P., Coutanceau, M.: Ecoulement instationnaire decolle dun fluide incompressible autour dun profil: une comparaison theorie – experience. AGARD CP 386, Paper 3 (1985)

Part XXIII

Gas-Kinetic BGK Schemes

A New High-Order Multidimensional Scheme Qibing Li, Kun Xu, and Song Fu

Abstract A third-order accurate multidimensional gas-kinetic BGK scheme is constructed through the high-order expansion of the distribution function and the high-order reconstruction of conservative variables. With several typical test cases the good performance of the new scheme is validated in both smooth flow and the flow with strong discontinuity. The theoretical validity for such an approach is due to the fact that the kinetic equation has no specific requirement on the smoothness of the initial data, as well as the simple particle transport mechanism and the inherent multidimensional characteristics on the microscopic level. The present study shows a new hierarchy to construct a high-order multidimensional method, and the NavierStokes flux function obtained from the present work can be adapted to many other high-order CFD methods.

1 Introduction The high-order numerical methods for the Navier-Stokes (NS) equations has attracted many researches due to its advantages in wide-range applications [1, 6]. The development of a solution under piecewise discontinuous high-order initial reconstruction for the NS equations directly is urgent, but difficult due to the mathematical inconsistency of the discontinuous initial data and the hyperbolic-parabolic nature of the NS equations. Furthermore, due to the lack of a multidimensional Riemann solution, it is also a great challenge to construct a genuinely multidimensional scheme. However, based on the gas-kinetic theory, it is easy to develop a generalized NS flow solver, BGK scheme [3, 7], due to the fact that the Boltzmann equation has no specific requirement on the smoothness of the initial data and the the simple particle transport mechanism at the microscopic level, including the coupling of the particle free transport and collisions, and the inherent multidimensional characteristics. The extension of the method from 2nd-order to high-order has been investigated, such as Q. Li (B) Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China e-mail: [email protected]


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the directional splitting method [4], or the multidimensional version for smooth flow [5]. This paper will further extend the method to construct a general high-order multidimensional gas-kinetic BGK method (HBGK-MD), suitable for not only smooth flow, but also flow with discontinuity.

2 A High-Order Multidimensional BGK Scheme 2.1 Fundamental of Gas-Kinetic BGK Scheme The construction of gas-kinetic BGK scheme [7] is briefly described as follows. First, the BGK-Boltzmann equation for two-dimensional (2-D) flow is written as ∂f g− f ∂f = + ui , ∂t ∂ xi τ

i = 1, 2 ,

(1)

where τ = μ/ p is the particle collision time. f = f (x, t, u, ξ ) is the gas distribution function, and g is the equilibrium state approached by f , assumed to be a Maxwellian distribution, g = ρ(2πRT )−(K +2)/2 e−λ(|u−U|

2 +ξ 2 )

,

(2)

where ξ 2 = ξ12 + ξ22 + . . . + ξ K2 represents the internal energy of particles with total number of degrees of freedom K = (4−2γ )/(γ −1). During the particle collisions, f and g satisfy the conservation constraint, (g − f )ψdΞ = 0 ,

ψ = (1, u, (|u|2 + ξ 2 )/2)T ,

(3)

at any point in space and time for the conservation of mass, momentum and energy. Here dΞ = du1 du2 dξ1 dξ2 . . . dξK is the volume element in the phase space. If the distribution function f is known, the macroscopic conservative quantities Q and the flux F can be obtained through the integration over the phase space Q = (ρ, ρU, ρV, ρ E) = T

f ψdΞ ,

F=

u f ψdΞ .

(4)

From Eqs. (1) and (3), the finite volume formulation of the BGK scheme is formed as n+1 n (Q∗ )lm = (Q∗ )lm +

1 Slm

,

t n +Δt

F∗ dtds

(5)

tn

where the computational cell is indexed by l and m with the area Slm and boundary s. The superscript “∗” represents the variable in the global coordinates. The flux F∗ is calculated through the coordinate transformation from that in the local coordinates

A New High-Order Multidimensional Scheme

631

F. For convenience, the calculation of F is presented through an example at a cell interface xs = (xl+1/2 , ym )T with xl+1/2 = 0 and −Δy/2 ≤ ym ≤ Δy/2. Now the question is how to solve the BGK equation (1) to obtain the gas distribution function f . To avoid the great difficulty of direct solving method, the gas-kinetic BGK scheme adopts a most ingenious method based on the ChapmannEnskog expansion and the integral solution of the BGK equation,

f (x, t, u, ξ ) =

1 τ

t

g(x , t , u, ξ )e−(t−t )/τ dt + e−t/τ f 0 (x − ut, u, ξ )

(6)

0

where x = x − u(t − t ) is the trajectory of a particle motion and f 0 is the initial gas distribution function at the beginning of each time step (t = 0). The ChapmanEnskog expansion is used to construct f 0 and g around the cell interface (l+1/2, m). Thus the time dependent distribution function f can be easily deduced and then the fluxes across the cell interface can be calculated with Eq. (4) and finally the conservative variables at the next time step can be calculated via the finite volume formulation (5). Details can be found in the corresponding reference. It should be noted that in the above-mensioned BGK method, f 0 and g can be constructed according to different purpose, such as that to approach high-order macro equation, i.e. BGK-Burnett [8], or to approximate NS equations with higher order accuracy [4, 5].

2.2 High-Order Multidimensional Scheme In order to develop a high-order accurate gas-kinetic BGK scheme, we can construct the high-order accurate initial distribution function f 0 and the equilibrium distribution g through the expansion to third-order in both spatial and temporal directions. The scheme for 1-D flow and 2-D flow with directional splitting method has been developed in our previous study [5]. Here, the genuinely multidimensional scheme can be constructed with the following f 0 and g, including both the normal and tangential slopes,

f 0 (x, 0, u, ξ ) = 1 + ail xi − τ ail u i + Al − τ ail Al + Cil xi

+ ail a lj + bil j (−τ u i x j + xi x j /2) (1 − H(x1 ))gl + 1 + air xi − τ air u i + Ar − τ air Ar + Cir xi

+ air a rj + birj −τ u i x j + xi x j /2 H(x1 )gr . g(x, t, u, ξ ) = g0 1 + ai xi + At + (ai a j + bi j )xi x j /2 + (ai A + Ci )xi t + (A2 + B )t 2 /2

(7)

(8)

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Q. Li et al.

where g0 is the initial local Maxwellians and H is the Heaviside function. The local terms ai , bi j , Ci , B and A are from the Taylor expansion of a Maxwellian and take the form, a = a (α) ψα , α = 1, 2, 3, 4, where all coefficients, a (α) , . . . , A(α) , are local constants from the first and second derivatives of g. These coefficients, as well as g0 are related to the reconstructed conservative variables Q and their slopes, which can be evaluated through the condition on Chapman-Enskog expansion, same as that in BGK-Burnett method [5, 8]. Then the distribution function at the cell interface can be deduced, f (xs , t, u, ξ ) = (1 − e−t/τ )g0 + (−τ + (τ + t)e−t/τ )ai u i g0 + (t − τ + τ e−t/τ )Ag0

+(1 − e−t/τ ) a22 + b22 x22 /2 g0 +(τ 2 − (t 2 /2 + τ t + τ 2 )e−t/τ )(ai a j + bi j )u i u j g0 +(t 2 /2 − τ t + τ 2 − τ 2 e−t/τ )(A2 + B )g0 +(2τ 2 − τ t − (2τ 2 + τ t)e−t/τ )(Ci + ai A)u i g0

+e−t/τ 1 − (t + τ )ail u i − τ Al + τ t Cil + ail Al u i

l x22 /2 + (τ t + t 2 /2) ail a lj + bil j u i u j H(u 1 )gl + (al )2 + b22 +e−t/τ 1 − (t + τ )air u i − τ Ar + τ t Cir + air Ar u i

r + (ar )2 + b22 x 22 /2 + (τ t + t 2 /2) air arj + birj u i u j (1 − H(u 1 ))g r . (9)

In the above equation, the variation of f along the tangential direction of the cell interface x2 is represented through the tangential slopes, such as a2 , b12 , b22 , C2 . However, the terms explicitly in proportion to x 2 are omitted, as the integration is zero. The terms containing x22 are retained, which is necessary for the scheme to achieve the third-order accuracy with only ONE integral point (the center of the cell interface). Furthermore, the above solution, or (6) allows the movement of particles in any direction. That is, the present high-order-accurate scheme simulates a multidimensional transport process across a cell interface. Thus it is a truely multidimensional scheme. For smooth flow field, the present method goes back to the previous simple scheme [4] (the term τt B is unnecessary at the level of NS equations), and the computational cost can be decreased remarkably. It should be noted that it is difficult to achieve high-order reconstruction of macro conservative variables for multidimensional flow. In the present study, the least-square method is adopted and the coefficients can be calculated in advance for only one time to decrease the computational cost. The 2nd-order PFGM limiter [9] ( p = 2) is used mostly for the direct reconstruction of conservative variables when the flow contains discontinuities.

3 Numerical Results Here we present some results computed by the newly developed HBGK-MD scheme. The first one is the inviscid isentropic vortex advection problem [10]. The computational domain is set to [−5, 5]2 divided by unform cells. The limiter is not

A New High-Order Multidimensional Scheme 100

633

L1 error L∞ error 3rd order

10–1

Error

10–2 10–3 10–4 10–5 10–6

0.1 Δx

0.2

0.3 0.4 0.5

Fig. 1 Errors in density vs. cell size at t = 10 for the isentropic vortex advection. The initial mean flow and the perturbation value are given by u = v = 1, p = 1, T = 1, and (δu, δv) = 2 2 (5/2π )e(1−r )/2 (−y, x), δT = −25(γ − 1)/(8γ π )e1−r

used in this case. Figure 1 shows the grid convergence of computed density, from which the third-order accuracy of the present method can be clearly observed. Figure 2 shows the velocity profiles at different streamwise locations for a boundary layer flow with Mach number 0.15 and Reynolds number Re = U∞ L/ν = 105 , where L = 100 is the length of a plane plate. The computational domain is chosen as [−40, 100] × [0, 50] and 120 × 30 grid cells are adopted with 40 × 30 cells locate ahead of the plate. The minimal cell sizes are Δx m = 0.1 and Δym = 0.07. One can see that the velocity distributions, not only for the streamwise component, but also for the transverse one, can be accurately predicted with only four cells, which shows the good performance of the present scheme in viscous flow. 1 1

+

+

+

+ 0.8

+

+

0.6

+ 0.4

+

+

+

0.2

x/L = 0.0247 x/L = 0.2625 x/L = 0.6239 Blasius

0.6

+ 0.4

+ 0.2

+ 0

0 ++ 2

4

y/(νx/U∞)1/2

6

8

+

+

+ 0

+

+

+

V/(νU∞/x)1/2

U/(νU∞/x)1/2

0.8

+

+

+

0

+

x/L = 0.0247 x/L = 0.2625 x/L = 0.6239 Blasius

+

2

4

y/(νx/U∞)1/2

Fig. 2 Velocity profiles at different streamwise locations in a boundary layer

6

8

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The two-dimensional shock tube problem [2] is calculated with the non-slip boundary at both two side walls and two end walls. Only half of the shock tube is considered divided by uniform cells, due to the symmetry of the flow in the vertical direction. The initial flow field is stationary, with sound speed a = 1 and a strong density jump at x = 0.5: ρ = 120 on the left and 1.2 on the right. As shown in Fig. 3, the complicated flow structures, such as the shock, boundary layer, vortex and their interactions, are well captured by the present scheme. Good grid convergence is achieved in the present study.

Fig. 3 Density contours for viscous shock tube problem with Re = 1,000 and Pr = 0.73 at time t = 1 (left: cell number 1,000 × 500, right: 2,000 × 1,000)

4 Conclusion In the present study, a high-order multidimensional gas-kinetic scheme is developed and validated with typical numerical tests. The constructed valuable NS flux function can be implemented to many high-order computational fluid dynamic methods. The comparison between the directional splitting scheme and a mutidimensional method, and the effect of the limiter for reconstruction require further study. Acknowledgements This work was supported by National Natural Science Foundation of China (Project No. 10872112).

References 1. Cockburn, B., Karniadakis, G.E., Shu, C.W. (eds.): Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, Berlin Heidelberg (2000) 2. Daru, V., Tenaud, C.: Numerical simulation of the viscous shock tube problem by using a high resolution monotonicity-preserving scheme. Comput. Fluids 38, 664–676 (2009) 3. Li, Q.B., Fu, S.: On the multidimensional gas-kinetic BGK scheme. J. Comput. Phys. 220, 532–548 (2006) 4. Li, Q.B., Fu, S.: A high-order accurate gas-kinetic BGK scheme. In: Choi, H., Choi, H.G., Yoo, J.Y.: (eds.) Computational Fluid Dynamics 2008. Springer-Verlag, Berlin Heidelberg (2009)

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5. Li, Q.B., Xu, K., Fu, S.: A high-order gas-kinetic Navier-Stokes flow solver. J. Comput. Phys. 229, 6715–6731 (2010) 6. Toro, E.F., Millington, R.C., Nejad, L.A.M.: Towards very high order Godunov schemes. In: Toro, E.F. (ed.) Godunov Methods: Theory and Applications, pp. 907–940. Kluwer/Plenum Academic Publishers, New York (2001) 7. Xu, K.: A gas-kinetic BGK scheme for the Navier-Stokes equations, and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171, 289–335 (2001) 8. Xu, K., Li, Z.: Microchannel flows in slip flow regime: BGK-Burnett solutions. J. Fluid Mech. 513, 87–110 (2004) 9. Yang, M., Wang, Z.J.: A parameter-free generalized moment limiter for high-order methods on unstructured grids. Adv. Appl. Math. Mech. 4, 451–480 (2009) 10. Yoon, S.H., Kim, C., Kim, K.H.: Multi-dimensional limiting process for three-dimensional flow physics analyses. J. Comput. Phys. 227, 6001–6043 (2008)

A Direct Boltzmann-BGK Equation Solver for Arbitrary Statistics Using the Conservation Element/Solution Element and Discrete Ordinate Method Bagus Putra Muljadi and Jaw-Yen Yang

Abstract This work presents a computational algorithm using discrete ordinate method with conservation element and solution element (CE/SE) scheme for solving the semiclassical Boltzmann equation with relaxation time approximation of Bhatnagar, Gross and Krook. The method is implemented on gases that obey arbitrary statistics distributions.

1 Introduction In the area of classical gas flows, the implementation of discrete ordinate method in the rarefied flow computations has been developed [3] and has been able to cover wide Knudsen number flow regimes. Under the same motivation, the present work is built using discrete ordinate method to describe the hydrodynamic properties of rarefied gases of all the three statistics. First, the discrete ordinate method is used to discretize the velocity space in the semiclassical Boltzmann-BGK equation into a set of equations in physical space with source terms. Second, the resulting equations can be treated as scalar hyperbolic conservation laws with stiff source terms whose evolution in space and time is modeled by an explicit, second-order CE/SE scheme developed by Chang [2].

2 Semiclassical Boltzmann-BGK Equation and Hydrodynamic Properties We first adopt the relaxation time concept of Bhatnagar, Gross and Krook, thus the semiclassical Boltzmann-BGK equation reads

B.P. Muljadi (B) Institute of Applied Mechanics, National Taiwan University, Taipei 10764, Taiwan e-mail: [email protected]


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p ∂f + · ∇x − ∇U (x, t) · ∇p ∂t m

f (p, x, t) =

δf δt

=− coll.

f − f (0) τ

(1)

see [1]. Here, U is the externally applied potential, m is the particle mass, p is particle momentum, τ is the relaxation time. The equilibrium distribution function for general statistics is expressed as f (0) (p, x, t) =

1 * +

2 z −1 ex p p − m u(x, t) /2mk B T (x, t) + θ

(2)

where m is the particle, u(x,t) is the mean velocity, T(x,t) is temperature, k B is the Boltzmann constant and z(x, t) = ex p(μ(x, t)/k B T (x, t)) is the fugacity, where μ is the chemical potential. In (2), θ = +1, denotes the Fermi-Dirac statistics, θ = −1, the Bose-Einstein statistics and θ = 0 denotes the Maxwell-Boltzmann statistics. The conservation laws of macroscopic properties in terms of macroscopic quantities i.e., number density n(x, t), momentum density nu(x, t), and energy density ε(x, t) are given by ∂n(x, t) + ∇x · j(x, t) = 0 ∂t dp p ∂mj(x, t) p f (p, x, t) = −n(x, t)∇x U (x, t) + ∇x · ∂t h3 m ∂ε(x, t) dp p p2 + ∇x · f (p, x, t) = −j(x, t) · ∇x U (x, t) ∂t h 3 m 2m

(3) (4) (5)

In one spatial dimension, the general distribution is given by f ( px , x, t) =

z −1 ex p

.

1

/ [ px − mu x ]2 /2mk B T (x, t) + θ

(6)

whereas n(x, t), j (x, t) and ε(x, t) are given by

dpx h dpx j (x, t) = h dpx ε(x, t) = h

n(x, t) =

f (0) ( px , x, t) =

Q1/2 (z) λ

px (0) f ( px , x, t) = n(x, t)u x (x, t) m Q3/2 (z) 1 px 2 (0) f ( px , x, t) = + mnu 2x 2m 2βλ 2

(7) (8) (9)

βh 2 Here, λ = 2π m is the thermal wavelength and β = 1/k B T (x, t). Quantum functions Qυ (z) of order υ are defined for Fermi-Dirac and Bose-Einstein statistics as

A Direct Boltzmann-BGK Equation Solver for Arbitrary Statistics

Fυ (z) ≡

1 Γ (υ)

1 Bυ (z) ≡ Γ (υ)

∞

∞

dx 0

∞

dx 0

639

l x υ−1 l−1 z (−1) ≈ lυ z −1 e x + 1

(10)

x υ−1 ≈ z −1 e x − 1

(11)

l=1 ∞ l=1

zl lυ

Here, Fυ (z) applies for Fermi-Dirac integral and Bυ (z) for Bose-Einstein’s, whereas Γ (υ) is gamma function. The normalized semiclassical Boltzmann-BGK equation is given by fˆ − fˆ(0) ˆ tˆ) ∂ fˆ(υˆ x , x, ˆ tˆ) ∂ fˆ(υˆ x , x, =− + υˆ x ∂ xˆ τˆ ∂ tˆ

(12)

From this part, our formulations are all considered normalized and we shall omit the “hat” sign for simplicity.

3 Application of Discrete Ordinate Method The application of the discrete ordinate method to Eq. (12) results in ∂ f σ (x, t) ∂ f σ (x, t) f σ − f σ (0) + υσ =− ∂t ∂x τ

(13)

with f σ and υσ represent the values of respectively f and υx evaluated at the discrete velocity points σ. Gauss-Hermite quadrature rule reads,

∞

−∞

N

2 2 ex p −υx Wσ ex p υσ 2 f (υσ ) ex p υx f (υx ) dυx ≈

(14)

σ=−N

where the discrete points υσ and weight Wσ can be found through √ 2n−1 n! π Wσ = 2 n [Hn−1 (υσ )]2

(15)

with n = 2N and υσ are the roots of the Hermite polynomial Hn (υ). The repeated Simpson’s rule is used at high temperatures instead. The macroscopic quantities can be described accordingly: e.g., normal density is given by

640


n(x, t) = =

∞

−∞ N

f (υx , x, t)eυx

Wξ

f σ e υσ

2

2

e−υx dυx 2

(16)

σ=−N

4 Numerical Method To solve the set of Eq. (13), a space-time CE/SE c − τ method with wiggle suppressing term is introduced. The two marching variables are described as

f σ nj

⎛ ⎞ n−1/2 n−1/2 − f σ (0) j Δt ⎝ f σ j ⎠= − − 2 τ n−1/2 n−1/2 1/2 (1 + c) f σ j−1/2 + (1 − c) f σ j+1/2 n−1/2 n−1/2 + (1 − c2 ) ( f σ x¯ ) j−1/2 − ( f σ x¯ ) j+1/2

(17)

where c = υσ (Δt/Δx) is the Courant number and ( fσ x¯ )nj = (Δx/4)( f σ x )nj is the normalized form of ( f σ x )nj . After f σ nj is known, another marching variable ( f σ x¯ )nj is determined by ( f σ x¯ )nj = (w−)nj ( fˆσx¯ −)nj + (w+)nj ( fˆσx¯ +)nj (w±)nj = W± ( fˆσx¯ −)nj , ( fˆσx¯ +)nj ; α

(18) (19)

where W∓ are functions given by W− (x − , x+ ; α) =

|x + |α |x − |α , W (x , x ; α) = , + − + |x − |α + |x+ |α |x − |α + |x + |α

(20)

for real variables x− , x+ and α ≥ 0. The arguments in the W∓ functions are specified as

n−1/2

( fˆσx¯ −)nj = ( f σ )nj − f σ − (2c + 1 − τc ) f σ x¯ j+1/2 /(1 + τc )

n−1/2 (21) ( fˆσx¯ +)nj = f σ − (2c − 1 + τc ) f σ x¯ j+1/2 − ( f σ )nj /(1 + τc ) Δt has to be less than τ . It also shall not violate the CFL stability condition, Δt S = c × (Δx/(υσ )max ). Thus, Δt = min(ΔtC , Δt S )

(22)

A Direct Boltzmann-BGK Equation Solver for Arbitrary Statistics

641

Numerical root finding methods are used to solve z(x, t), which is the root of Ψ1 (z) = ε −

Q3/2 4π

n Q1/2

3

1 − nu 2x 2

(23)

5 Computational Results The results are shown in terms of 1-D shock tube problems. CE/SE is used with α = 4. (i) Mesh refinement test. The condition applied to Fermion gas is (n l , u l , Tl ) = (0.557, 0, 1) and (nr , u r , Tr ) = (0.341, 0, 0.6). These correspond to zl = 0.4

(a)

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Fig. 1 Case (i) ( : 100 grids; : 200 grids; solid line : 400 grids); case (ii) ( : Euler; : τ = 0.0001; : τ = 0.001; : τ = 0.01; : τ = 0.1); case (iii) ( : Euler; : Kn = 0.0001; : Kn = 0.001; : Kn = 0.01; : Kn = 0.1); case (iv) ( : MB; : FD; ♦ : BE). (a) case (i), (b) case (ii), (c) case (iii) and (d) case (iv)

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and zr = 0.3. Three Uniform grid systems are used with 100 , 200 and 400 cells. CFL = 0.5 and τ = 0.001. The quadrature used is Gauss-Hermite with 20 abscissas. (ii) Varying relaxation times. The initial condition applied to Fermion gas at both sides of the tube are (nl , u l , Tl ) = (0.724, 0, 4.38) and (n r , u r , Tr ) = (0.589, 0, 8.97) which correspond to zl = 0.225 and zr = 0.12. τ ranges from 0.1 to 0.0001. (iii) Varying Knudsen √ numbers. The relaxation time will vary with Kn 5 π K n Q3/2 (z) number according to τ = . Initial condition of case (ii) are used 8nT (1−χ ) Q5/2 (z) with Knudsen number ranging from 0.1 to 0.0001. (iv) Test on high temperatures. In this test, the temperatures at both sides are set as: Tl = 24.38 and Tr = 22.97 whereas fugacity and velocity are kept fixed. The results can be seen in Fig. 1. In case (i) the density profiles converge to the highest grid points. The same behavior is observed in Case (ii) and (iii) where those with the highest Kn and τ converge to Euler limit. Case (iv) illustrates how the three statistics recover to the classical limit at high temperatures.

6 Concluding Remarks A direct algorithm that applies discrete ordinate method and CE/SE for solving semiclassical Boltzmann-BGK transport equation for particles of all statistics is constructed. Different aspects of the numerical method are tested. The feasibility of this algorithm have been illustrated without much major constraints. Acknowledgements This work is done under the auspices of National Science Council, TAIWAN through grants NSC-99-2922-I-606-002.

References 1. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94(3), 511–525 (1954). doi:10.1103/PhysRev.94.511 2. Chang, S.C., To, W.M.: A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element. NASA TM (104495) (1991). doi:10.1103/PhysRevE.79.056708 3. Yang, J.Y., Huang, J.C.: Rarefied flow computations using nonlinear model boltzmann equations. J. Comput. Phys. 120(2), 323–339 (1995). doi:10.1006/jcph.1995.1168. http://www. sciencedirect.com/science/article/B6WHY-45NJJFW-1J/2/275dde267bf1a59e6a3cf435fa6e68ce

Part XXIV

Extreme Flows

Space-Time Convergence Analysis of a Dual-Time Stepping Method for Simulating Ignition Overpressure Waves Jeffrey A. Housman, Michael F. Barad, Cetin C. Kiris, and Dochan Kwak

Abstract High-fidelity time-accurate simulations of the launch environment are an important part of the successful launch of new and existing space vehicles. The capability to predict certain aspects of the launch environment, such as ignition overpressure (IOP) waves, is paramount to mission success. Dual-time stepping implicit methods can provide accurate results in a timely manner. To determine the necessary solver parameters for accurate three-dimensional simulations, space-time convergence analysis on a simplified two-dimensional problem related to IOP wave prediction is considered. Using two separate numerical methods, sensitivity analysis was performed over a large parameter space including physical-time step, number of dual-time sub-iterations, and grid resolution.

1 Introduction Upon ignition of a rocket propulsion system, large-magnitude ignition overpressure (IOP) waves propagate from the nozzle with potentially damaging effects. The IOP waves are generated during the buildup of thrust, in which mass is suddenly injected from the nozzle into the confined volume of the flame trench under the launch platform. The additional mass displaces air in the trench, causing a piston-like action in which compression waves travel up and down beneath the Mobile Launch Platform (MLP). Reflections of the IOP waves can travel back towards the launch vehicle with the potential to affect its structural integrity. Predicting the magnitude, frequency, and direction of the IOP waves is critical to understanding the launch environment. For more details on the ignition overpressure phenomenon see Jones [3]. High-fidelity Computational Fluid Dynamics (CFD) simulations have been established as a central component in the safety assessment of the launch environment for new and existing spacecraft, see Kiris et al. [4]. The ability to predict specific launch environment phenomena such as ignition overpressure (IOP) waves is critical to the successful launch of a vehicle. One method to provide accurate J.A. Housman (B) ELORET Corp., Sunnyvale, CA 94086, USA e-mail: [email protected]


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results, using a reasonable amount of computational resources, is a dual-time stepping method. The dual-time stepping method considered here is an implicit numerical method for unsteady flows in which a pseudo-time process is embedded into each physical-time step, see Rogers et al. [5]. This procedure allows efficient steady-state convergence techniques to be applied at each physical-time step. It also allows the use of relatively large physical-time steps compared to conventional approximate Newton methods. While time-accurate CFD simulations offer a powerful prediction tool for modeling unsteady phenomenon such as IOP waves, it is often difficult to quantify the error in the computed results. For example, use of excessively large time steps combined with incomplete convergence of the sub-iteration procedure may generate non-physical and spurious solutions that are difficult to detect. Currently, no well-established theory exists on necessary conditions (i.e. number of subiterations, residual convergence, etc.) to maintain a specified physical-time accuracy while minimizing the computational cost per time step. The sub-iteration procedure could be performed until convergence to machine precision is achieved, but this is too costly for many engineering applications.

2 Problem Definition In this paper, we analyze the space-time convergence behavior of the dual-time stepping method applied to a two-dimensional model problem simulating IOP waves. The IOP waves are generated from the ignition of a single nozzle over a 45◦ plate. Figure 1 shows the schematic of the nozzle profile and 45◦ plate with 14 pressure extraction point locations indicated in red. Due to lack of space, only results for

Fig. 1 Two-dimensional IOP model geometry with pressure extraction point locations

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location 6 are reported in this paper, but indings were consistent across locations. This model problem was jointly defined by the Japan Aerospace Exploration Agency (JAXA) and the National Aeronautics and Space Administration (NASA) through a collaborative agreement. In this work, space-time sensitivity analysis is performed over a large parameter space including physical-time step, number of dual-time sub-iterations, and grid resolution. Error in the simulated results are compared using traditional convergence analysis techniques including both global and local quantities of interest. Comparisons of unsteady pressure history at specified locations in the domain, as well as time integrated functionals are included in the analysis. Compressible Euler computations were performed using two separate time-derivative preconditioned flow solvers. The first research code, an overset structured grid solver, uses alternating line implicit relaxation (Overset). The second code uses block-structured adaptively refined Cartesian grids and point implicit relaxation (AMR). The Overset methodology is described in Housman et al. [2], while the AMR code utilizes the Chombo numerical library, see Barad et al. [1] and references therein. Three grid resolutions were considered: Δx = 10.5/2,048 (Coarse), Δx = 10.5/4,096 (Medium), and Δx = 10.5/8,192 (Fine). Here Δx represents the minimum grid spacing used in the region of interest near the pressure extraction points and the nozzle. For each grid resolution, 42 unsteady simulations were performed using a parameter space of physical-time steps and dual-time sub-iterations (N SU B). The physical-time step Δt = C F LΔx/cref , where cref = 340 ms−1 is the reference sound speed and C F L = 1, 2, 5, 10, 50, and 100. For each C F L, N SU B = 5, 10, 20, 50, 100, 200, and 400 were used. For the overset code a constant pseudo-time C F L pseudo = 10 was used, while the AMR code used a local pseudo-time C F L pseudo = 1.

3 Results Figure 2 shows a time sequence of gauge pressure using the overset code. The sequence shows the initial pressure wave as it is released from the nozzle (t = 0.0036 s), interacts with the 45◦ plate (t = 0.0072 s), develops the Mach cone structure (t = 0.0108 s), the reflected wave interacts with the nozzle (t = 0.0144 s), the reflected waves become more solitary and radially propagates away from the jet (t > 0.0180 s). Figure 3 shows a numerical Schlieren from the AMR code, at time t = 0.0180 (s). Adaptive mesh refinement (AMR) was specified to track pressure and temperature gradients. In this sequence of plots, two additional levels of mesh refinement track the gradients, but are hidden to better visualize the Schlieren. This figure highlights: the Mach cone diamond structure, the shock structures along the 45◦ plate, the reflected waves as they travel away from the jet, and vortical folding in the shear layer as the jet propagates down the plate.

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Fig. 2 Time sequence of gauge pressure (psig) using the overset code. X- and Y-axis labels are in meters


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Fig. 3 Numerical Schlieren (|∇ρ|) at t = 0.0180 (s) using the AMR code. Black boxes indicate computational cells on coarser AMR grid levels. X- and Y-axis labels are in meters

Figure 4 presents the average sub-iteration convergence for the (a) overset and (b) AMR simulations. This average was taken over the entire duration of the simulation. Trends in data show that an increase in N SU B improves the subiteration convergece, while an increase in physical time C F L degrades the it, as expected. For C F L > 5, it was found that convergence was weak even for 400

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Fig. 4 Average sub-iteration convergence as a function of N SU B/C F L, (a) overset and (b) AMR

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sub-iterations. Similar weak convergence is observed for N SU B < 20. For these runs, a 20 ≥ N SU B/C F L ≥ 40 leads to sufficient sub-iteration convergence. Figure 5 plots time series of gauge pressure at sensor location 6. Plots (a) and (b) show sub-iteration convergence of the time series for C F L = 1. Plots (c) and (d) show time step convergence of the time series for N SU B = 400. Both codes converge to a similar time series for N SU B ≥ 20 or C F L ≤ 10. AMR (b) appears to perform better than overset (a) for N SU B = 10, similarly AMR (d) initially performs better than overset (c) for C F L = 50. These are likely due to local time stepping used in the AMR code to enhance the sub-iteration convergence. Figure 6 presents a code-to-code comparison of the space-time convergence of gauge pressure at location 6. In plot (a) the time series is shown for both codes at each of the three grid resolutions for C F L = 1 and N SU B = 400. As the mesh resolution increases, higher frequencies are observed after the initial pressure pulse (t > 0.012). Plot (b) shows that the root-mean-squared (RMS) gauge pressure

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Fig. 6 Code-to-code comparison of space-time convergence of gauge pressure at location 6: (a) full time-series, (b) root-mean-squared of time-series, (c) min value of time-series, (d) max value of time-series

converges to similar values for both codes. The minimum and maximum pressures over the integration time also converge to similar values as shown in (c) and (d). Although the maximum gauge pressure (d) is sub-iteration converged at each grid resolution, it is not yet space-time converged for either code. This suggests that further refinement is necessary if the maximum pressure is considered important. Additionally, viscous effects should be included to damp non-physical frequencies.

4 Conclusion A space-time convergence analysis was performed on a two-dimensional ignition overpressure model problem. Results from this analysis quantitatively show that the ratio of number of sub-iterations to the physical-time CFL is approximately

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independent of mesh resolution. This ratio also provides a good metric for solution accuracy for a given space-time resolution. If time integrated (e.g. RMS) functionals are of interest, then coarser space-time resolutions may be acceptable. If maximum or minimum pressure values are important, then care must be taken to insure that higher frequency data is either resolved or filtered for space-time convergence. This analysis will be repeated to include viscous effects, which may help to damp higher frequency wave content.

References 1. Barad, M. F., Colella, P., Schladow, S. G.: An adaptive cut-cell method for environmental fluid mechanics. Int. J. Numer. Meth. Fluids, 60(5), 473–514 (2009) 2. Housman, J., Kiris, C., Hafez, M.: Time-derivative preconditioning methods for multicomponent flows – Part II: Two-dimensional applications. J. of Appl. Mech., 76, (2009) 3. Jones, J. H.: Scaling of ignition startup pressure transients in rocket systems as applied to the space shuttle overpressure phenomenon. APL JANNAF 13th Plume Technology Meeting, 1, 371–392 (1982) 4. Kiris, C., Housman, J., Gusman, M., Chan, W., Kwak, D. Time-accurate computational analysis of the flame trench applications. 21st International Conference on Parallel Computational Fluid Dynamics, pp. 37–41 (2009) 5. Rogers S. E., Kwak D., Kiris, C.: Steady and unsteady solutions of the incompressible NavierStokes Equations. AIAA J. 29, 603–610 (1991)

Numerical Solution of Maximum-Entropy-Based Hyperbolic Moment Closures for the Prediction of Viscous Heat-Conducting Gaseous Flows James G. McDonald and Clinton P.T. Groth

Abstract The use of hyperbolic high-order moment closures for the numerical prediction of gas flows is considered. These closures provide transport equations for an extended set of fluid-mechanic properties that can account for the evolution of gases both in and out of local thermodynamic equilibrium. Also, the first-order nature of moment equations allows for the prediction of viscous or heat-transfer effects without the need for the computation of second derivatives. Numerical solutions to first-order hyperbolic equations are less sensitive to grid irregularities which often result from adaptive-mesh-refinement or embedded-boundary techniques. In addition, the lower requirements on the order of the derivatives also means that numerical schemes for moment equations can in general gain an extra order of spatial accuracy for the same reconstruction stencil when compared to equations requiring the use of second-order derivatives. This work examines the practical use of high-order maximum-entropy moment closures for the prediction of viscous, heat-conducting gas flows. As a test problem, shock-structure calculations are shown for a wide range of shock Mach numbers.

1 Introduction The use of fully hyperbolic high-order moment closures for the numerical prediction of gas flows provides several immediate advantages. These closures lead to transport equations for an extended set of fluid-mechanic properties that can account for the evolution of gases both in and out of local thermodynamic equilibrium. This is in contrast to traditional fluid-mechanic equations, such as the Euler and NavierStokes equations, which are only valid in or near local equilibrium. Deviations from equilibrium are common for micron-scale flows; highly rarefied flows; and flows with extreme gradients, as are encountered within shock waves. In addition to modelling advantages, hyperbolic moment closures present many numerical advantages. The first-order nature of moment equations allows for the prediction of viscous or J.G. McDonald (B) University of Toronto Institute for Aerospace Studies, Toronto, ON, Canada M3H 5T6 e-mail: [email protected]


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heat-transfer effects without the need for the computation of second derivatives; again, this is in contrast to the Navier-Stokes equations which rely on elliptic operators for the prediction of these phenomena. Numerical solutions to first-order hyperbolic equations are less sensitive to grid irregularities which often result from adaptive-mesh-refinement or embedded-boundary techniques [3, 7]. The lower requirements on the order of the derivatives which must be approximated during a computation also means that numerical schemes for moment equations can in general gain an extra order of spacial accuracy for the same reconstruction stencil when compared to equations requiring the use of second-order derivatives. It is noted that many numerical schemes are well-suited for the computation of solutions to hyperbolic equations (such as Godunov-type finite-volume or discontinuous-Galerkin finite-element schemes) while other methods are more appropriate for the solution of elliptic or parabolic equations. However, schemes which handle the different natures of terms in conservation equations of mixed type with equal elegance have proven to be more difficult to develop. For all these reasons, the purely hyperbolic nature of moment equations is a highly desirable characteristic [3, 6–8].

2 Moment Closures Moment closures arise from the field of gaskinetic theory. In this field, the particle nature of gases is not ignored, as is done in more traditional fluid dynamics. However, rather than attempt a representation of the evolution of each individual gas molecule, probability density functions, F (xi , vi , t), are used to describe the number of molecules at a particular location and time which have a particular velocity. Macroscopic properties of the gas are found by computing velocity moments of the distribution function. For example, the density, ρ, and bulk or average velocity, 555 u i , of a gas are computed as ρ(xi , t) = mF (xi , vi , t) d3 vi = mF and ∞ mvi F u i (xi , t) = m F , were m is the molecular mass. The particle random velocity, ci = vi −u i , can now be defined. The evolution of F is described by the Boltzmann equation, ∂F ∂F δF = + vi . ∂t ∂ xi δt

(1)

The term on the right hand side of the equation, δδtF , is the collision integral. This term represents the effects of intermolecular collisions on the distribution function. The result is that (1) represents a six-dimensional integro-differential equation and its numerical solution represents an excessive numerical expense for most situations. Fortunately, for many practical applications, it is only the moments of the distribution which are of interest. The evolution of these macroscopic quantities can be found by taking velocity moments of (1). This leads to Maxwell’s equation of change, ∂ ∂ m M(vi )F + mvi M(vi )F = [M(vi )F ] . ∂t ∂ xi

(2)

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Here M(vi ) is the velocity weight of interest and [MF ] is the effect of intermolecular collisions on its corresponding moment. It is clear in (2) that the time evolution of a moment will always be dependent on the spacial divergence of a moment that is one order higher in terms of the particle velocity. In general, the use of Maxwell’s equation of change therefore would lead to an infinite set of moment equations. One technique for obtaining a closed finite system of moment equations is to restrict the distribution function to some assumed form that is a function of several free parameters which are determined by requiring the distribution function to be consistent with the chosen moments of interest. This technique was originally pioneered by Grad [2], who restricted the distribution function to be a polynomial expansion about the equilibrium Maxwell-Boltzmann distribution. Unfortunately, this choice leads to moment equations that are only hyperbolic for a region surrounding local equilibrium, outside of which flow problems become ill-posed. Recently it has been found that there are many advantages to obtaining closure by choosing the distribution function which is consistent with the moments of interest and has the maximum entropy. A systematic hierarchy of closures based on this assumption has been proposed by Levermore [5]. This hierarchy provides moment equations which have been shown to be globally hyperbolic whenever the underlying entropy maximization problem is solvable. Numerical simulations of gas flows using a lower-order member of this hierarchy (the Gaussian closure) have also been examined [3, 6, 7, 9] and show considerable promise for the prediction of continuum and non-equilibrium gas behaviour. The Gaussian closure, however, is deficient in that it has no treatment of heat transfer and is therefore unsuitable for any situation in which heat transfer plays a significant role. Higher-order members of the Levermore hierarchy do offer a treatment for heat transfer, however there are two issues with these closures. Firstly, there is no closed expression for the closing fluxes as a function of the closure moments. This leads to the requirement that an entropy maximization problem must be solved every time a flux must be determined. This maximization problem depends on moments of a distribution function which must be integrated numerically over a domain which stretches from negative to positive infinity in all directions. Secondly, the maximumentropy moment system is only well defined for moment values for which the underlying entropy maximization problem can be solved. It has been shown previously [4] that for all members of this hierarchy of sufficiently high order to offer a treatment of heat flux, there exist physically realistic regions for which this maximization has no solution and a distribution function cannot be realized. In these regions, the maximum-entropy closure technique breaks down and corresponding moment equations do not exist.

2.1 One-Dimensional Maximum-Entropy Moment Closure In order to investigate the practical application of maximum-entropy based moment closures, a simple one-dimensional version of the moment equations is considered. In this case, “one-dimensional” refers to the fact that gas particles can only have velocities in one space direction. The simplest one-dimensional moment closure

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that is a member of the Levermore hierarchy and offers a treatment for heat trans

T fer is a five-moment system with primitive solution vector W = ρ, u, p, q, r . A simplified relaxation-time collision operator is used [1]. The resulting moment equations are ∂ρ ∂t

∂ ∂ x (ρu) = 0 , 2 ∂ ∂ ∂t (ρu) + ∂ x ρu + p = 0 , ∂ 3 2 ∂ ∂t ρu + p + ∂ x ρu + 3up + q = 0 , ∂ 4 3 q ∂ 2 ∂t ρu + 3up + q + ∂ x ρu + 6u p + 4uq + r = − τ , ∂ ∂ 4 2 5 3 2 ∂t ρu + 6u p + 4uq + r + ∂ x ρu + 10u p + 10u q + 5ur + s =

2 − τ1 4uq + r − 3 pρ .

+

(3) (4) (5) (6)

(7)

Here the pressure is p = mc2 F , heat transfer is q = mc3 F , and higher order 5 4 moments are r = mc F and s = mc F . It is the moment s which is not a member of the solution vector, and must therefore be determined from a closure relation. It can be shown that the maximum-entropy distribution function for this system can be non-dimensionalized such that ρˆ = pˆ = 1 and uˆ = 0. The nondimensionalized closing relation is therefore only a function of the non-dimensional qˆ and rˆ [4]. It can also be shown that the only states that can be realized by a positive distribution function are contained on the plain rˆ ≥ 1 + qˆ 2 [4]. Incidentally, for this moment closure, the region or physically realizable states for which the entropy-maximization problem has no solution that was proven to exist by Junk is the line qˆ = 0 and rˆ > 3. This is particularly troubling as local equilibrium is at the point qˆ = 0 and rˆ = 3; in fact, for all high-order maximum-entropy closures, local equilibrium lies on the boundary of the region of non-realizability.

2.2 Maximum-Entropy Surface Fit As stated earlier, one of the major stumbling blocks to the adoption of maximumentropy-based moment closures is the lack of a closed-form expression for closing fluxes. It will be shown that a simple surface fit can provide an adequate approximation to the true maximum-entropy closure above, (3), (4), (5), (6) and (7). Firstly, it should be noted that the line defining the region of physical realizability, rˆ = 1 + qˆ 2 , represents a distribution function that is comprised of two delta functions. On this line the closing relationship can be found analytically and ˆ Next, realizing that the region of realizability is parabolic, it seems is sˆ = qˆ 3 + 2q. sensible to parametrize the space using a parabolic transformation as follows rˆ =

2qˆ 2 +3−σ σ

with

0≤σ≤2.

(8)


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For this mapping, lines of constant σ are parabolas and σ is the distance down from local equilibrium, rˆ = 3, that these lines intersect the rˆ axis. These parabolas have curvatures that increase from σ = 2, where the parabola coincides with the limit of physical realizability, to σ = 0 where the parabola collapses to the line qˆ = 0 and rˆ ≥ 3, thus covering the entire realizable region. It was found through numerical experimentation, that along the lines of constant σ the moment sˆ can be well approximated by a cubic function of qˆ as sˆ = d3 (σ)qˆ 3 + d1 (σ)q. The functions d3 (σ) and d1 (σ) can be fit by first numerically finding finite difference approximations to these derivatives along the line qˆ = 0 and 1 ≤ rˆ ≤ 3. These data points were then fit using standard fitting software; it was found that they are well approximated by the functionals d1 = a + bσ + cσ2 +dσ3 + eσ4 + f σ5 + gσ6

d3 =

a + bσ + cσ2 + dσ3 1 + eσ + f σ2 + gσ3

with a b c d e f g

with

= 9.9679007422678190 = −9.234367231975216 = 8.2142492688404296 = −7.372320367163680 = 4.3920303941514343 = −1.452821303578764 = 0.2006200057926356

a b c d e f g

= −20,840.93761193234 = 7,937.3772948278038 = 405.05250560053173 = −329.3827765656151 = −1,077.797102997202 = −3,072.303291055466 = 1,056.0890741355661

Figure 1a, b show the closing flux of the maximum entropy system as well as the surface fit shown above. The relative error is plotted in Fig. 1c. It can be seen that away from the line on which the maximum-entropy distribution does not exist and the predicted flux is singular, the fit is actually quite good. In practise, it is quite advantageous that the fit does not closely approximate the singularity and rather transitions smoothly across the rˆ axis.

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3 Numerical Calculations of Shock Structures As a preliminary investigation into the behaviour of the fitted moment closure, shock waves of Mach numbers 2, 4, and 8 are considered. Comparisons are made to high-resolution simulation of the kinetic equation (1), with the same relaxation-time collision operator as was used in the moment equations [1]. In this way, all of the error is due to the closure approximation and not the collision operator. Comparison is also made to the equivalent Navier-Stokes-like equations for this one-dimensional gas. Figure 2a–c show normalized density profiles for shocks with a Mach number of 2, 4, and 8 respectively, while normalized heat transfer is shown in Figure 2d–f. It can be seen that agreement between the moment equations and the BGK equation is generally very good, far better than the Navier-Stokes-like equations. As with all hyperbolic systems, a discontinuity appears in the moment solution when the incoming flow speed exceeds the maximum wavespeed in the system. In this case, however, the size of the jump is very small. Figure 1c shows the orbits traced by these shock profiles in the q-ˆ ˆ r plane. All shocks begin at equilibrium, jump to a non-equilibrium state across the discontinuity, and return smoothly to equilibrium. It can be seen that in all cases the area of largest error in the fit is avoided. Numerical experience suggests that the closure remains hyperbolic throughout the explored regimes.

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4 Conclusions Although maximum-entropy-based moment closures are known to have several apparent disadvantages, including a lack of a closed-form expression for closing fluxes and regions of non-realizability, this paper has demonstrated that these difficulties can be handled in practise, at least for some closures. For the onedimensional five-moment system studied, the closing flux can be conveniently fit with a simplified expression and hyperbolicity seems to be preserved. Unlike with many other moment closures, the discontinuities present within high-Machnumber shock calculations are small and may be considered acceptable for many applications.

References 1. Bhatnagar, P.L., Gross, E.P., Krook, M.: A model for collision processes in gases. i. small amplitude processes in charged and neutral one-component systems. Physical Rev. 94(3), 511–525 (1954) 2. Grad, H.: On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2, 331–407 (1949) 3. Groth, C.P.T., McDonald, J.G.: Towards physically realizable and hyperbolic moment closures for kinetic theory. Cont. Mech. Therm. (2009). doi:10.1007/s00161-009-0125-1 4. Junk, M.: Domain of definition of Levermore’s five-moment system. J. Stat. Phys. 93(5/6), 1143–1167 (1998) 5. Levermore, C.D.: Moment closure hierarchies for kinetic theories. J. Stat. Phys. 83, 1021–1065 (1996) 6. McDonald, J.G., Groth, C.P.T.: Numerical modeling of micron-scale flows using the Gaussian moment closure. AIAA Paper 2005-5035 (2005) 7. McDonald, J.G., Groth, C.P.T., Sachdev, J.S.: Application of Gaussian moment closure to micro-scale flows with moving and embedded boundaries. AIAAJ (2011, submitted) 8. Suzuki, Y., Khieu, L., van Leer, B.: CFD by first order PDEs. Cont. Mech. Therm. (2009). doi:10.1007/s00161-009-0124-2 9. Suzuki, Y., van Leer, B.: Application of the 10-moment model to MEMS flows. AIAA Paper 2005-1398 (2005)

Supercritical-Fluid Flow Simulations Across Critical Point Satoru Yamamoto, Takashi Furusawa, and Ryo Matsuzawa

Abstract A numerical method for simulating supercritical-fluid flows across the critical point is introduced. The present numerical method is based on the preconditioning method developed by the authors and mathematical models of thermophysical properties developed by Kyushu University. The thermophysical properties such as density, viscosity, heat conductivity and the derived values are rapidly changed across the critical point. Especially the isobaric specific heat has a maximum peak value. The present method can simulate both low- and high-speed flows with the anomalous properties across the critical point accurately. As numerical examples, two different supercritical carbon-dioxide flows across the critical points are calculated and the calculated results are introduced.

1 Introduction We know a number of substances such as water, carbon-dioxide, oxygen, nitrogen, and so on. These substances have their own thermophysical properties. These properties are changed according to bulk conditions. Unfortunately, all the existing CFD solvers assuming ideal gas or incompressible fluids cannot calculate such real flow problems. We proposed a numerical method [4] for flows of arbitrary substance in arbitrary conditions. In this chapter, our method is extended to a method for simulating both low-speed and high-speed supercritical-fluid flows across the critical point. This method is based on the preconditioning method developed by the authors [3] and is fully coupled with the database of thermophysical properties of fluids, PROPATH, developed by Kyushu University [9]. The present preconditioning method is based on a compressible flow solver and we derived a preconditioned flux-vector splitting(PFVS) scheme [3]. The PFVS enables us to calculate both low-speed and highspeed flows. S. Yamamoto (B) Department of Computer and Mathematical Sciences, Tohoku University, Sendai 980-8579, Japan e-mail: [email protected]


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In PROPATH, thermophysical models for 48 substances are programmed in wide-range pressure and temperature conditions. All of these models have been verified and validated as the most accurate model in chemical engineering. For examples, equation of state(EOS) for carbon-dioxide employed in PROPATH was standardized by International Union of Pure and Applied Chemistry(IUPAC) [8]. As EOS for water, a unified model at the International Association for the Properties of Water and Steam(IAPWS) IF-97 [5] is employed. PROPATH can cover all the states except for solid, that is gas, liquid and supercritical fluid. Each model is defined as a function of pressure and temperature. All the thermophysical properties used in the present computational code are referred from PROPATH as a function. The present method has been already applied to the calculation for several flows of supercritical water and carbon dioxide across the critical point. In this chapter, two typical calculated results of supercritical carbon-dioxide flows across critical point are introduced.

2 Numerical Methods The fundamental equations used in this study are based on the preconditioned twodimensional compressible Navier–Stokes equations in curvilinear coordinates. The system of the equations is written in a vector form as ˆ ∂ Fi ∂ Fvi ˆ ˆ = Γ ∂Q + + = 0 (i = 1, 2) Γ ∂ Q/∂t + L( Q) ∂t ∂ξi ∂ξi

(1)

where ⎡

θ ⎢ θ u1 Γ =⎢ ⎣ θ u2 θ h − (1 − ρh p )

0 ρ 0 ρu 1

⎤ 0 ρT 0 ρT u 1 ⎥ ⎥ ρ ρT u 2 ⎦ ρu 2 ρT h + ρh T

h = (e + p)/ρ, ρT = ∂ρ/∂ T , h T = ∂h/∂ T and h p = ∂h/∂ p. θ is the preconditioning parameter defined by θ = 1/Ur2 − ρT (1 − ρhp )/ρh T . The switching parameter Ur was defined by Weiss and Smith [6]. The numerical flux (Fi )*+1/2 for Fi in Eq. (1) defined at the interface between the control volume * and * + 1 in each coordinate i (i = 1, 2) can be written in a flux-vector splitting form as (Fi )*+1/2 = Fi+ *+1/2 + Fi− *+1/2

L = Aˆ + + Aˆ − Qˆ *+1/2 i

*+1/2

i

*+1/2

R Qˆ *+1/2

(2)

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The superscripts ± indicate the sign of characteristic speeds. Fi± are the numerical flux-vectors, consisting of only positive or negative characteristic speeds. Aˆ i± are the preconditioned Jacobian matrices, consisting of only positive or negative characteristic speeds. Qˆ L and Qˆ R are the unknown vectors extrapolated from the

and right directions. The preconditioned flux-vector splitting form for left ± M ˆ Qˆ *+1/2 derived in this study is given by Ai *+1/2

Aˆ i±

*+1/2

Qˆ M = Γ L i−1 Λi± L i

*+1/2

Qˆ M

(3)

The superscript M is replaced by L or R. The final derived form of Eq. (3) was presented by Yamamoto [3]. The present method employs the implicit method based on LU-SGS scheme [2]. This scheme is modified to the preconditioned LU-SGS scheme [3] as Γ DΔ Qˆ ∗ = R H S + Δt G + (Δ Qˆ ∗ ) ˆ Δ Qˆ = Δ Qˆ ∗ − Γ −1 D −1 Δt G − (Δ Q)

(4) (5)

where D is the diagonal matrix approximated by the spectral radius of the preconditioned Jacobian matrices and R H S is the vector of explicit time-marching residues for Eq. (1). G ± consists of non-diagonal time-derivative fluxes. The detailed form of Eqs. (4) and (5) was also derived by Yamamoto [3]. All the mathematical models for thermophysical properties programmed in the present computational code are referred to from PROPATH as an external function. Most of those models are formed by a polynomial derived from the existing theoretical equations or experimental data. In this chapter, only the models for carbon dioxide are introduced. The EOS for carbon dioxide is the one standardized by the International Union of Pure and Applied Chemistry(IUPAC) [8]. The EOS model is defined by ⎡ p = ρ RT ⎣1 + ω

Ji 9

⎤ ai j (τ − 1) j (ω − 1)i ⎦

(6)

i=0 j=0

where ω = ρ/ρ ∗ and τ = T ∗ /T . The coefficients ai j and the number Ji are defined in [8]. Actually, in this case, ρ ∗ = 468 [kg m−3 ] and T ∗ = 304.21 [K]. Thermophysical properties, such as isometric and isobaric specific heats, can be derived using Eq. (6) and theoretically defined by Cv =

ρ 0

T ρ2

∂2 p ∂T 2

ρ

dρ + Cvideal , C p = Cv +

C videal is the isometric specific heat for ideal gas.

T (∂ p/∂ T )2ρ ρ 2 (∂ p/∂ρ)T

(7)

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The partial derivatives derived from Eqs. (6) and (7) can be calculated from the combination of thermophysical properties using Bridgman’s table, as defined in PROPATH. Molecular viscosity and thermal conductivity for carbon dioxide are also modeled by a polynomial equation. Since the thermophysical properties are modeled by a polynomial function, the calculation generally uses a lot of CPU time. In this study, all thermophysical properties used are interpolated from the look-up tables in which the values are discretely calculated once from the polynomial function programmed in PROPATH.

3 Numerical Examples The calculated results of two-dimensional natural convection of carbon dioxide between two parallel plates in a vertical direction is first introduced. The numerical results have been already reported by Davis [1]. A case assuming a Rayleigh number of 104 in atmospheric pressure and temperature conditions is calculated. The bulk pressure is 0.1 [MPa]. The temperatures on the left and right plates are 303 and 313 [K], respectively. This case is further extended to natural convection in a supercritical-pressure condition. The pressure is increased to 8.0 [MPa]. In this condition, the phase change between supercritical fluid and liquid occurs at an intermediate value of the temperatures between 303 and 313 [K]. Figure 1a, b show the calculated density distributions at the pressure of 8.0 [MPa] and that of 0.1 [MPa], respectively. The density contours in Fig. 1a are quite different from those in Fig. 1b. The magnitude of density in Fig. 1a is two orders higher than that in Fig. 1b because carbon dioxide is in a supercritical fluid or liquid state; in contrast, the carbon dioxide is gas in Fig. 1b. The magnitude of the density difference in Fig. 1a is also far different from that in Fig. 1b. The density is almost constant in Fig. 1b, while about 2.5 times the density difference is observed in Fig. 1a. Figure 2a, b show the calculated distributions of isobaric specific heat at the

(a) Fig. 1 Calculated density distributions. (a) 8.0 MPa, (b) 0.1 MPa

(b)


(a)

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(b)

Fig. 2 Calculated distributions of isobaric specific heat. (a) 8.0 MPa, (b) 0.1 MPa

pressure of 8.0 [MPa] and that of 0.1 [MPa], respectively. Drastic distributions are obtained in Fig. 2a. A zonal region where the isobaric specific heat has a maximum peak is captured in Fig. 2a. Usually, the value is almost constant in a gas, as obtained in Fig. 2b. The maximum peak value in Fig. 2a is approximately seven times higher than the minimum value in Fig. 2b. As the second problem, the calculated results of a supercritical carbon- dioxide flow through a nozzle with free-jet expansion is introduced. This flow has been used to fabricate nanoscale particles in the process, the so-called RESS process. In this chapter, the whole flow field from the nozzle region to the expansion chamber is calculated assuming an axisymmetric flow. Then, the thermophysical properties are changed from those in a supercritical condition to an atmospheric condition across the critical point through the nozzle. The flow is also accelerated from a static condition to a supersonic speed. The nozzle minimum diameter D at the nozzle throat is D = 5 × 10−5 [m]. This nozzle is a capillary nozzle system. The inlet temperature T0 is 350 [K]. The outlet pressure p∞ is 0.1013 [MPa]. Figure 3 shows the calculated Mach number distributions assuming the pressure ratio p0 / p∞ = 10.0, where p0 is the inlet pressure, which is in a subcritical pressure

Fig. 3 Calculated Mach number distributions

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CASE 1 CASE 2 CASE 3

ρ /ρ0

0.8 0.6 0.4 0.2 0.0 –2.0

–1.0

0.0

1.0

2.0

3.0

x/D

Fig. 4 Calculated density ratio distributions on the x-axis

condition. The distributions indicates that the shock system including Mach disk, a barrel shock, a reflection shock, and a jet boundary, is clearly captured by the present method. Crist [7] reported that the distance to Mach disk is only influenced by the pressure ratio p0 / p∞ and is not dependent on the difference of substance and the temperature. This case (CASE1) is extended to be in supercritical pressure and temperature conditions. As additional cases, the outlet pressure p∞ is varied to 1.0 and 2.0 [MPa] fixing the pressure ratio p0 / p∞ = 10.0. These cases are corresponding to CASE 2 and CASE 3. Then, CASE 2 and CASE 3 are a supercritical pressure and temperature condition at the inlet and the condition is certainly changed to a gas condition through the nozzle throat, while CASE 1 is totally in a gas condition. Figure 4 shows the calculated distributions of the density ratio ρ/ρ0 on the axis from the front of the nozzle throat to the downstream of the Mach disk. Although the same pressure ratio, discrepancies in the density ratio appear when the carbon dioxide fluid streams in the nozzle throat.

4 Conclusion A numerical method based on the preconditioning method and mathematical models for thermophysical properties of several substances programmed in PROPATH was developed. Typical two calculated results of supercritical carbon dioxide flows across the critical point were introduced. Both results indicate that thermophysical properties in supercritical carbon-dioxide flow across critical point are quite different from those in gas or liquid flows. The accurate prediction of the thermophysical properties will be absolutely required for simulating such flow problems.


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References 1. Angus, S., et al.: International Thermodynamic Table of the Fluid State-3 Carbondioxide, IUPAC, 3 (1976) 2. A Program Package for Thermophysical Properties of Fluids(PROPATH), Ver.12.1, PROPATH GROUP 3. Crist, S., Sherman, P.M., Glass, D.R.: AIAA J. 4, 68–71 (1966) 4. de Vahl Davis, G.: Int. J. Num. Meth. Fluids 3, 249–264 (1983) 5. Japan Society of Mechanical Engineers.: 1999 JSME Steam Tables (1999) 6. Weiss, J.M., Smith, W.A.: AIAA J. 33, 2050–2056 (1995) 7. Yamamoto, S.: J. Comp. Phys. 207, 240–260 (2005) 8. Yamamoto, S., Furusawa, T.: Proceedings of the 6th International Conference on CFD-Seoul, pp. 545–550. Springer, Seoul (2008) 9. Yoon, S., Jameson, A.: AIAA J. 26, 1025–1026 (1988)

Numerical Model for the Analysis of the Thermal-Hydraulic Behaviors in the Calandria Based Reactor Mula Venkata Ramana Reddy, S.D. Ravi, P.S. Kulkarni, and N.K.S. Rajan

Abstract Fuel channel integrity in a reactor depends on the coolability of the moderator which serves as an ultimate heat sink under transient normal working conditions. Hence a thorough understanding of the behavior of the moderator is very important in the safe design of a reactor. CFD investigations are carried out to study the peak thermal energy estimation in a typical calandria using industry standard CFD code. Numerical flow computations and experimental studies are carried out for a calandria embedded with a matrix of tubes working together as a reactor. Numerical investigations are carried out in the calandria reactor considering 480 fuel channels, with tangential inlets 14◦ inclined to the vertical axis, and with varied number of outlets at the bottom portion of the reactor model. The outlets are vertically downward, and inclined at 30◦ to the vertical axis. The work is carried to study the flow pattern and the associated temperature distribution. The computations are made for simulations of flow and convective heat transfer for assigned near-to working conditions with different moderator injection rates and reacting heat fluxes. The numerical results estimated isothermally and using 3D RANS equations are validated against experimental results. A set of experiments on a specially designed scale model for automated flow visualization are conducted over a range of flows and simulation parameters. The work assumes significance for the design considerations of the reactors and for detailed and critical parametric studies for optimization of the geometry considered.

1 Introduction The present world demands an environmental friendly power generation for which nuclear power plant is a suitable alternative. The nuclear power is associated with a scare regarding its safety during operation [2]. Hence there is a need to analyze flow pattern and heat transfer in calandria or any heat exchanger used in these systems. M.V.R. Reddy (B) Computational Mechanics Laboratory, JATP, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India e-mail: [email protected]


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The study of thermal and hydraulic behaviours using complex geometries and configurations is considered significant in industrial applications. The study gained momentum to have safety designs and optimised performances. This study conceives high relevance in industries. The numerical analysis needs to be supplemented by experimental data for reliability. Heat transfer, fluid flow and boiling are important phenomena in many industrial applications such as nuclear reactor system [3, 5]. In case of nuclear reactor complete information of the moderator flow and the temperature distribution within the calandria are significant and vital in providing feed back for safe design. CFD results obtained from different mesh models and various grid sizes are validated against experimental results. From the comparison, the right numerical model is selected for thermal analysis inside the calandria. In the process using CFD tool three different case studies (i.e., single outlet, two outlets and three outlets) of actual prototype for different mass flow rates and heat flux are carried out. The work assumes significance for preliminary design of the reactors and for detailed and critical parametric analysis that prove to be expensive without CFD tools.

2 Calandria Model Calandria flow analysis has significant effect on safe operation of reactors especially pressurized heavy water reactors (PHWR). Figure 1 shows the 3D- view of the calandria model with two inlets and two outlets. Figure 2 shows the geometry considered for CFD simulation and Fig. 3 shows the hexahedron – mesh model for corresponding geometric modeling. An experimental model of constituting 69 fuel channels in the core region, with 21 mm of square pitch is considered for numerical and experimental analysis of the behavior around the channels. The results are in close agreement with each other

Fig. 1 3-D view of the calandria model

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Fig. 2 Calandria model considered for simulation

Fig. 3 Structured grid model

and thus, prompt one to go ahead with an idea of numerically analyzing a bigger (480 channels) model on the similar lines.

3 Boundary and Initial Conditions A 3D-RANS code having upwind implicit scheme and k−ω approach for turbulence is used for the numerical solution. The Reynolds-Averaged Navier–tokes Equations are solved for steady, compressible viscous flow. The governing equations used are the conventional standard set of equations that include: Continuity equation: ∂U j =0 ∂x j

(1)

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Momentum equation:

p ∂ ∂ ∂ τi j + ρu i u j − ρU i U j = − ρU i + ∂t ∂x j ∂ xi ∂x j

(2)

Energy equation:

p ∂ ∂ Q j + ρu j h ρU j h = − ρh + ∂t ∂x j ∂x j

(3)

The boundary and initial conditions used include (a) no slip, impermeable and adiabatic walls; (b) At inlet and outlet ports, mass flow rate conditions based on incoming and outgoing incompressible fluid are imposed. The mass flow rate at inlet and outlet are chosen to be equal. (c) For boundary conditions, axi- symmetry is considered. Steady state, stationary domain, non-buoyancy, low intensity turbulence and static temperature of 300 K are used. Water at 25 ◦ C as domain fluid and reference pressure of 1 atmosphere is used. Diminishing residual criteria of the variables is used for the convergence with a limit of RMS residuals falling below 10−4 .

4 Computational and Experimental Results The flow pattern and the temperature distribution have been captured spatially. The Reynolds number is calculated based on mass flow rate of the flow. The nonisothermal analysis is made with an assumption that the fuel channel surfaces are giving out uniform heat flux. The isothermal flow distribution is used as an initial approximation to the flow field for saving the computation time. This initialization is also analogous to the typical start-up procedures of the reactor. It is observed that the flow pattern and the velocity profiles in the non-isothermal case are in close agreement with those of the isothermal case indicating that the flow is controlled dominantly by the momentum than the buoyancy [1]. Figure 4 shows the comparison of the experimental results to be in close agreement with those of numerical results. Figure 5 depicts temperature distributions. Figure 6 shows hot spots

Experimental

Numerical

Fig. 4 Flow pattern comparision

Experimental

Numerical


103 MW single outlet

103 MW two outlets

Fig. 5 Temperature distribution for 621 kg s

1200 MW, single outlet

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−1

1200 MW, two outlets

1200 MW, three outlets

Fig. 6 Hot spot mapping for 621 kg s−1 mass flow

Fig. 7 Flow pattern for single outlet, two outlets and three outlets

mapping. The streamline plots are shown in Fig. 7. The effect of heat load on temperature is shown in Fig. 8. Figures 9 and 10 represent the influence of mass flow rate on fuel channel wall temperature and that on outlet temperature respectively. The trend agrees with the natural prediction. Figure 11 depicts experimental set up for flow visualization [4].

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Fig. 8 Effect of heat load on temperature along horizontal cross section

Fig. 9 Effect of Inlet Mass flow rate on fuel channel wall temperature

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Fig. 10 Effect of Inlet Mass flow rate on outlet temperature

Fig. 11 Different parts of experimental setup with calandria model

5 Conclusions The results are in coherence with the natural trend, that the frequency of hot spots decreases as the number of outlets increases and temperature dissipation increases as inlet mass flow injection rate increases.

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The numerically obtained results provide an estimate of the tolerance bands of safe working limits for the heat dissipation for different working conditions, by virtue of capturing the hot spots in the calandria. The work assumes significance for preliminary design considerations of the reactors and for detailed and critical parametric analysis that prove to be expensive without CFD tools. The approach can be used for optimization of the geometrical relocations of the in lets and outlets in combination of the different arrangements in mounting of the fuel channels. Acknowledgements The first author expresses his thanks to the authorities, IISc-Bangalore, India, for their finacial support towards meeting registration and partial accomodation charges. The first author also expresses his sincere thanks to M/s Boening company USA for their finacial support through “Boeing Air Travel Felloship” to meet air travel and living expenses.

References 1. Adak, A.K., Rao, I.S., Srivastava, V.K., Tewari, P.K.: Nuclear desalination by waste heat utilization in an advanced heavy water reactor. Int. J. Nuclear Desalination 2(3), 234–243 (2007) 2. Doria, F.J.: Thermal hydraulic safety characteristics of CANDU reactors. CANDU Safety Presentations. Shanghai, 1–23. http://canteach.candu.org/library/19990104.pdf (2002) 3. Kim, M., Yu, S.-O., Kim, H.-J.: Analysis on fluid flow and heat transfer inside Calandria vessel of CANDU-6 using CFD. J. Nucl. Eng. Design 236(11), 1155–1164 (2006) 4. Patankar, S.V.: Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC (1980) 5. Tupake, R.S., Kulkarni, P.S., Rajan, N.K.S.: Numerical Analysis of Heat and Mass Transfer in a Calandria Based Reactor. In: Proceedings of the 5th Asian CFD Conference, Bussan, Korea (2003)

Part XXV

Deformable Bodies

Seamless Virtual Boundary Method for Complicated Incompressible Flow Simulation Hidetoshi Nishida

Abstract In this chapter, the seamless virtual boundary method for the complicated incompressible flow simulation is presented. In order to satisfy the velocity conditions on the virtual boundary points, the forcing term is added not only on the grid points near the boundary but also on the grid points inside the boundary in the seamless virtual boundary method, so that the smooth physical quantities can be obtained. In the case with heat transfer, the additional heat flux term is considered in the energy equation. The flow around a heated circular cylinder for the heat transfer problem and the flow around a swimming fish model for the complicated moving boundary problem are considered. The results show that the present seamless virtual boundary method is very promising for the numerical simulation of incompressible flows with the heat transfer, complicated geometry and moving boundary.

1 Introduction For the practical flow simulations, the boundary fitted coordinates (BFC) is usually adopted. This BFC approach has the high adaptability to the boundary configuration. However, in the complicated flow geometry, it is difficult to generate the computational grid. Moreover, in the BFC approach, it is necessary that the governing equations are transformed from the physical plane to the computational plane, so that the transformed governing equations have more terms than the original equations in the Cartesian coordinates. Therefore, the computational effort is larger than the Cartesian grid approach. Then, in recent years, the Cartesian grid approach is highlighted again for the numerical flow simulations. In the Cartesian grid approaches, especially, the immersed boundary method is applied to many simulations of incompressible flow. One of this immersed boundary method is the virtual boundary method [2, 6]. In order to satisfy the velocity conditions on the (virtual) boundary points, the virtual boundary method requires H. Nishida (B) Department of Mechanical and System Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan e-mail: [email protected]


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the external forcing term added to the momentum equations. This forcing term is usually estimated by two ways, i.e., feedback [6] and direct [2] procedures. Also, in both forcing term estimations, the forcing term is added to the grid points only near the boundary. Therefore, the unphysical oscillations near the boundary appear in the pressure field. Nishida and Sasao [4] proposed the seamless virtual boundary method. In this method, the forcing term is added not only on the grid points near the boundary but also in the region inside the boundary, so that the unphysical pressure oscillations can be removed. In this chapter, we try to apply the seamless virtual boundary method to the complicated flow simulation with heat transfer and moving boundary.

2 Seamless Virtual Boundary Method 2.1 Governing Equations In the virtual boundary method, the velocity conditions on the virtual boundary are satisfied by adding the forcing term to the momentum equations. Then, the incompressible viscous flow with heat transfer is governed by the continuity equation, the incompressible Navier – Stokes equations, and the energy equation. These equations can be written in nondimensional form by ∂u i = 0, ∂ xi ∂u i ∂p 1 ∂ 2ui ∂u i +uj =− + + Gi , ∂t ∂x j ∂ xi Re ∂ x j ∂ x j ∂T 1 ∂T ∂2T = + GT , +uj ∂t ∂x j Re Pr ∂ x j ∂ x j

(1) (2) (3)

where u i , p, and T denote the velocity component, the pressure, and the temperature. Re(= UL/ν) denotes the Reynolds number and Pr (= α/ν) is the Prandtl number. U , L, α, and ν are the reference velocity, the reference length, the thermal diffusivity, and the kinematic viscosity, respectively. The last terms in Eqs. (2) and (3), G i and G T , denote the additional forcing term and the additional heat flux term, respectively.

2.2 Seamless Virtual Boundary Method In order to estimate the additional forcing term in the governing equations, G i , there are mainly two ways, that is, the feedback and direct forcing term estimations. In this chapter, the direct forcing term estimation shown in Fig. 1 is adopted. We

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Fig. 1 Direct forcing estimation

explain in 2D but the extension to 3D is straightforward. For the forward Euler time integration, the forcing term can be determined by n n+1 U − u in ∂u i ∂p 1 ∂ 2ui Gi = u j + − + i , ∂x j ∂ xi Re ∂ x j ∂ x j Δt n+1

(4)

where U i denotes the interpolated velocity by using U and u i+1 shown in Fig. 1. U is the boundary velocity, e.g., U = 0 for stationary solid media and U = Umove for moving boundary, where Umove is the moving velocity. Namely, the external force is specified as the velocity components at next time step satisfy the relation, n+1 u in+1 = U i . In this forcing term estimation, the grid points added forcing term are restricted near the boundary. Then, the pressure distributions near the boundary show unphysical oscillations. In order to remove aforementioned unphysical oscillations near the boundary, the seamless virtual boundary method [4] was proposed. In the seamless virtual boundary method, the forcing term is added not only near the boundary but also in the region inside the boundary shown in Fig. 2. On the grid points near the boundary, the additional forcing term is estimated by the same procedure as the usual direct forcing term estimation, Eq. (4). In the region inside

(a)

(b)

Fig. 2 Grid points added forcing term. (a) Usual direct forcing, (b) seamless forcing

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the boundary, the forcing term is determined by satisfying the relation, U i = Uvb , where Uvb is the specified velocity, e.g., Uvb = 0 in stationary solid media. In the energy equation, the typical conditions for temperature are the isothermal and adiabatic conditions. For the isothermal boundary, the additional heat flux term can be estimated in the same manner as the momentum equations. n n+1 ∂T T − Tn 1 ∂2T − + , GT = u j ∂x j Re Pr ∂ x j ∂ x j Δt

(5)

n+1

where T denotes the interpolated temperature by Tvb and Ti+1 , where Tvb is the temperature on the isothermal boundary. For the adiabatic boundary, the boundary temperature Tvb is estimated by the adiabatic condition, ∂ T /∂n = 0, where n is the normal direction. By using Tvb and Ti+1 , the additional heat flux term G T can be computed from Eq. (5). Also, in the energy equation, this heat flux term is added not only near the boundary but also in the region inside the boundary in the seamless virtual boundary method.

2.3 Numerical Method The incompressible Navier – Stokes equations (2) and energy equation (3) are solved by the second order finite difference method on the collocated grid arrangement. The convective terms are discretized by the second order fully conservative finite difference method [3]. The diffusion and pressure terms are discretized by the usual second order centered finite difference method. For the time integration, the fractional step approach based on the two-step Runge-Kutta scheme [5] is applied. For the usual incompressible Navier – Stokes equations, the two-step Runge-Kutta scheme is written by

u in+1 = (1−β3 )u in +β3 u in−1 +Δt β1 Fin − ∇ p n + β2 Fin−1 − ∇ p n−1 , (6) where β1 , β2 , and β3 are the parameters of two-step Runge-Kutta scheme and Fi denotes the convective and diffusion terms. In this chapter, the parameters are set as β1 = 9/8, β2 = −7/8, β3 = −3/4 in order to satisfy the second order accuracy in time. The energy equation is integrated in the same manner. The fractional step approach based on the two-step Runge-Kutta scheme can be written by

u i∗ = (1 − β3 )u in + β3 u in−1 + Δt β1 Fin + β2 Fin−1 − ∇ p n−1 ,

(7)

u in+1 = u i∗ − Δt β1 ∇ pn − G i ,

(8)

where u i∗ denotes the fractional step velocity. The resulting pressure equation is solved by the SOR method.


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3 Numerical Results 3.1 Validation of Seamless Virtual Boundary Method In order to validate the present seamless virtual boundary method, the flow around a circular cylinder with Re = 40 is considered. The pressure distributions near the boundary obtained by the usual virtual boundary method and the present method are shown in Fig. 3. Also, Fig. 4 shows the pressure along the center line. It is very clear that the present seamless virtual boundary method can remove the unphysical oscillations near the boundary in comparison with the usual solution. In Fig. 4,

Fig. 3 Comparison of pressure field 2.0

:VBM1 :VBM2 :VBM3

Pressure

1.5

fluid

solid

fluid

1.0

0.5 9.5

10.0

11.0

X

Fig. 4 Pressure along the center line

11.5

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the usual pressure distribution (VBM1,2) has the pressure jump at the boundary. On the contrary, the present pressure profile (VBM3) near the boundary is very smooth. Then, it is confirmed that the present method can give the precise pressure field.

3.2 Flow Around a Heated Circular Cylinder In order to validate the estimation of additional heat flux term, the flow around a heated circular cylinder is considered. Figure 5 shows the snapshots of temperature with Re = 200. The vortex shedding is formed and the heat is transfered to the downstream by the vortex. In the close-up view, the temperature inside a circular cylinder is constant, so that the isothermal condition is satisfied. Also, the time averaged local Nusselt number on a circular cylinder with Re = 218 is shown in Fig. 6. The present result is in good agreement with the other solutions, i.e., Tanno et al. [7], Xi et al. [8], and Eckert et al. [1]. Then, it is confirmed that the present estimation of additional heat flux term is appropriate.

(a)

(b)

Fig. 5 Snapshot of temperature field (Re = 200). (a) Whole view, (b) close-up view

3.3 Flow Around a Fish Model Finally, the flow around a stationary and a swimming fish models at low Reynolds number is considered. The 5 leveled hierarchical grid with about 3.8 million grid points is used. The surface shape is generated by the triangular polygon. The swimming fish is controlled by 2D rotation on four rotating axes with 100 frames per one period. On the virtual boundary and inside the boundary, the moving velocity condition is imposed. Figure 7 shows the pressure distributions around a stationary fish model with Re = 100. The smooth pressure field can be obtained near the fish surface and in the fins. Figure 8 shows the pressure distributions with Re = 100 on x − y plane. At the tail fin, it is confirmed that the high pressure region appears in the


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Fig. 6 Time averaged local Nusselt number (Re = 218)

(a)

(b)

Fig. 7 Pressure profile of a stationary model. (a) y = 5.5D plane, (b) z = 5.4D plane

motion direction side and the low pressure region appears in the motion direction reverse side. The second invariant of velocity gradient tensor near a swimming fish is shown in Fig. 9. The comparatively small eddy with the motion of the abdomen, hip and back fins can be observed clearly. Also, it is confirmed that the thrust force is arising in this swimming case. Then, the present seamless virtual boundary method gives the physically appropriate solution for the complicated flow analysis with moving boundary.

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(a)

(b)

Fig. 8 Pressure profile of a swimming model. (a) 30/100 frame, (b) 80/100 frame

(a)

(b)

(c)

(d)

Fig. 9 Iso-surface of second invariant of velocity gradient tensor. (a) 5/100 frame, (b) 35/100 frame, (c) 65/100 frame, (d) 95/100 frame

4 Concluding Remarks In this chapter, the seamless virtual boundary method is presented. The present method can remove the unphysical oscillations near the boundary, so that the precise pressure field can be obtained. The present method can be applied not only to the heat transfer problem but also to the moving boundary problem, successfully. Then, it is concluded that the present seamless virtual boundary method is very smart computational tool for complicated incompressible flow simulations with moving boundary and heat transfer.

References 1. Eckert, E.R.G., Soehngen, E.: Distribution of heat-transfer coefficients around circular cylinders in crossflow at Reynolds numbers from 20 to 500. Trans. ASME 74, 343–347 (1952) 2. Fadlun, E.A., Verzicco, R., Orlandi, P., Mohd-Yosof, J.: Combined immersed-boundary finitedifference methods for three-dimensional complex simulations. J. Comput. Phys. 161, 35–60 (2000)


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3. Morinishi, Y., Lund, T.S., Vasilyev, O.V., Moin, P.: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143(90), 90–124 (1998) 4. Nishida, H., Sasao, K.: Incompressible flow simulations using virtual boundary method with new direct forcing terms estimation. In: Deconinck, H., Dick, E. (eds.) Computational Fluid Dynamics 2006, pp. 371–376. Springer, Heidelberg (2009) 5. Renaut, R.A.: Two-step Runge-Kutta method and hyperbolic partial differential equation, Math. Compt. 55(192), 563–579 (1990) 6. Saiki, E.M., Biringen, S.: Numerical simulation of a cylinder in uniform flow: Application of a virtual boundary method. J. Comput. Phys. 123, 450–465 (1996) 7. Tanno, I., Morinishi, K., Matsuno, K., Nishida, H.: Validation of virtual flux method for forced convection flows. Trans. JSME Ser. B, 72(714), 217–224 (2006). (in Japanese). 8. Xi, G., Torikoshi, K., Kawabata, K., Suzuki, K. 2006 Numerical analysis of unsteady flow and heat transfer around bodies using a compound grid system. Trans. JSME Ser. B 61(585), 1796–1803 (in Japanese) (2006)

Modeling, Simulation and Control of Fish-Like Swimming Michel Bergmann and Angelo Iollo

Abstract We consider modeling and simulation of two-dimensional flows past deformable bodies. The incompressible Navier – Stokes equations are discretized in space on a fixed Cartesian mesh and the bodies are taken into account using a penalization method. The interfaces between the bodies and the fluid are tracked using a level-set description. It is thus possible to simulate several bodies freely evolving in the fluid. The application considered is fish-like swimming and the propulsion efficiency for different swimming modes is investigated.

1 Introduction The modeling and simulation of fish-like swimming is of interest in life sciences as well as in engineering applications. Understanding the mechanics of swimming can help clarifying some aspects of the evolution and of the physiology of aquatic organisms. In engineering, the study and optimization of aquatic locomotion can improve the design of underwater vehicles having superior maneuvering capabilities. The aim of this study is to present a simulation technique that is an extension to moving objects of an existing method [1] highlighted onto different swimming modes. In particular we are concerned with the power required for swimming as a function of the swimming modes and eventually of the relative position of the swimmers.

2 Modeling and Simulation of Flow Around Self Propelled Bodies Our aim is to study a two dimensional incompressible flow around Ns moving and deformable bodies. The entire domain is noted Ω = Ω f ∪ Ωi , i = 1, . . . , Ns , where domain Ω f is filled with the fluid and the domains Ωi denote the bodies with M. Bergmann (B) INRIA Bordeaux Sud Ouest, Team MC2, Bordeaux F-33000, France; Institut de Mathmatiques de Bordeaux, UMR 5251, Bordeaux F-33000, France e-mail: [email protected]


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boundaries ∂Ωi . The bodies around which the flow is computed are modeled using the so-called penalization method in which the bodies are considered as porous media with a very small permeability K ( 1. Let χi denote the characteristic function for the i th body, so that χi (x, t) = 1 if x ∈ Ωi and χi (x, t) = 0 elsewhere. In the limit of K → 0 it can be shown [1], that the solution obtained by the following method converges to the solution of the Navier – Stokes equations where ui is the velocity of body i. On Ω we have: Ns ∂u 1 χi (ui − u) in Ω, + (u · ∇)u = −∇ p + Δu + K ∂t Re

(1a)

∇ · u = 0 in Ω,

(1b)

1=1

with initial conditions in Ω and boundaries conditions on ∂Ω. Since we do not have to impose any boundary conditions onto the body, Eqs. (1a) and (1b) can be discretized onto Cartesian grids, thus avoiding the use of body fitted meshes. Equations (1a) and (1b) are discretized in space using finite differences onto a Cartesian grid (centered second order for the pressure and viscous terms, and upwind third order for the convective terms) and using a fractional step method [3] in time. The body velocity ui (in the penalization term of eq (1a)) can be decomposed into ui and a deformation B ui velocity. While the deformation a translation ui , a rotation C velocity is imposed, the translation and rotation velocities are computed from aerodynamic forces and torques respectively (using Newton’s laws). Since we do not have a direct access to the body boundary (use of Cartesian mesh), we use a volume integral form to compute forces and torques as reported in [4]. For later convenience we introduce a level set function φi that represents the signed distance function of a given point to ∂Ωi [5]. It is negative if x ∈ Ωi and positive elsewhere. The level set φi = 0 is the interface between the fluid and the bodies immersed in the flow. The characteristic function can be deduced from i − th distance function as χi = 1 − H (φi ) where H is the Heaviside function. Once the bodies velocities are computed, the position of the bodies denoted by the characteristic function φi can be obtained from the advection equation ∂φi + (u · ∇)φi = 0, ∂t

i = 1, . . . , Ns .

(1c)

The system (1) is solved thanks to a parallel code written using the PETSc1 libraries. Since we use Cartesian meshes the parallelism is very easy and efficient, the numerical code shows a good scalability. Since the classical penalization technique described below is of first order in space the gradients are not consistent. We thus focus on the improvement of the penalization order. The test case we chose is the Green – Taylor vortex with 1 Portable, Extensible Toolkit for Scientific Computation, see http://www.mcs.anl.gov/petsc/ petsc-as/

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analytical solution (see [2] for more details about Green – Taylor vortex). At the center of the computation domain we impose a disk on that we penalized the equaana denotes the analyttion. We first chose u1 (x, y, t) = uana 1 (x, y, t) where u1 ical solution and we obtain the second order of the numerical scheme. We then impose u1 (x, y, t) = u1 (x, y, t) where (x, y) denote the coordinates of the closest boundary point to the point (x, y). We obtain the first order. If we use level set information with the following correction u1 (x, y, t) = u1 (x, y, t) − φi (∂ ui (x, y, t)/∂n)n−1 , we obtain a second order penalization. All these numerical results are reported in Fig. 1. –2

analytic penalization 1st order penalization 2nd order penalization

–4

Error

–6 –8 –10 –12 –14 –16 –4.5

–4

–3.5

–3

–2.5

Δx

Fig. 1 Improvement of the penalization order

3 Application to Fish Like Swimming After having presented the method to generate both fish shape and swimming law (§3.1), we will focus on the different swimming modes (§3.2). We will then perform an energy study including burst and coast swimming (§3.3) and fish schooling (§3.4).

3.1 Generation of the Fish Shape and Swimming Law The steady body of a fish can be approximated in 2D using a Karman Trefftz trans

from z = n

1+ ζ1

n

n

+ 1− ζ1

n

n 1 1+ ζ − 1− ζ1

(see Fig. 2). The imposed backbone deformation is

computed as y(x, t) = a(x) sin(2π(x/λ + f t)), where the curve envelope is given

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y

rc

x=

ηc 0

η

0

x

ζ space

z space

(a)

(b)

Fig. 2 Sketch of the Karman – Trefftz transform. (a) Original shape, (b) steady shape y

y

y y(x, t)

0

s= x

(a)

centering

cercle with radius r

s

x

0 x

(b)

x

0 translation and rotation

(c)

Fig. 3 Sketch of swim and maneuvers shape. (a) Swimming, (b) maneuvering, (c) real motion

by a(x) = c1 x + c2 x 2 (see Fig. 3). This motion is defined by a constant phase speed c p = λ f , where λ and f denote the wavelength and the frequency of the oscillations respectively. The fish shape is defined by n, ηc and the length *, the swimming law is defined by λ, f , c1 and c2 . Fixing c1 and c2 one can determine the maximal tail amplitude and vice versa. For maneuvering purpose a mean curvature can be added to the swimming deformation. Is this study we present several examples of fish like swimming problem. It is well known now that fishes generate a reverse von Karman street leading to a propulsive jet like behavior. Energy savings is one of the most desirable aim for fish swimming onto long distance. To get fair comparison, in the examples shown hereafter, all fishes swim at the same average velocity (imposed by regulating the tail amplitude).

3.2 Influence of the Steady Periodic Swim Mode A convergence study (not presented here) for the number of grid point have showed that 128 points per unit length (Δx = Δy = 1/128) is a good compromise between precision and CPU computation time. This is why the following simulations have been performed using such a grid, and the time step is taken to be Δt = 0.2Δx. The von Karman street generated by a thunniform and anguiliform fish are presented in Fig. 4. The von Karman streets are different for each fish mode (thunniform, carangiform, subcarangiform and anguiliform). It has been numerically demonstrated that the thunniform fishes are the fastest ones. In term of energy saving, we suppose

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Fig. 4 Vorticity representation of the wakes generated by thunniform (top) and anguiliform (bottom) at Re = 103

that all fish swim at the same velocity onto the same distance. It has then been demonstrated that the thunniform spends the less energy compared to the others.

3.3 Influence of the Intermittent Swimming (Burst and Coast) A swimming period is equal to a period when the fish swims form the minimal velocity Ui to a maximal velocity U f plus a period when the fish glides from U f to Ui . A good choice of U f and Ui can lead to an energy saving up to 40% in comparison with the steady periodic swim. A vorticity representation of the wakes generated by a burst and coast fish at Re = 103 is given in Fig. 5.

Fig. 5 Vorticity representation of the wakes generated by a burst and coast fish at Re = 103

3.4 Influence of Fish Schooling It is also demonstrated that appropriate organization of a fish school allows to save a great amount of energy [6]. Indeed, we show numerically that a three fish school can save 15% of energy if one fish swim in the midway wake generating by two fishes swimming in a frontal way.

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4 Conclusions We have presented a method to model and simulate self propelled bodies. The method is based on Cartesian meshes, penalization to take into account the bodies and level set functions to track the interface between the fluid and the bodies. The Navier – Stokes equations are solved using a fractional step method and the body motions are computed from the Newton’s laws (forces and torques). The capability of this method to simulate self propelled bodies is highlighted for fish like swimming. Some energetic investigations have been done. We are now working onto the three dimensional extension of the method, including Eulerian elasticity for fluidstructure interactions.

References 1. Angot, P., Bruneau, C., Fabrie, P.: A penalization method to take into account obstacles in a incompressible flow. Num. Math. 81(4), 497–520 (1999) 2. Bergmann, M.: Optimisation arodynamique par rduction de modle POD et contrle optimal. Application au sillage laminaire d’un cylindre circulaire., Ph.D. thesis, Institut National Polytechnique de Lorraine, Nancy, France (2004) 3. Chorin, A.: Numerical solution of the Navier-Stokes equations. Math. Comp. 22, 745–762 (1968) 4. Noca, F.: On the evaluation of time-dependent fluid-dynamic forces on bluff bodies, Ph.D. thesis, California Institute of Technology, Pasadena, CA (1997) 5. Sethian, J.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, Cambridge (1999) 6. Weihs, W.: Hydrodynamics of fish schooling. Nature 241, 290–291 (1973)

Multi-Level Quasi-Newton Methods for the Partitioned Simulation of Fluid-Structure Interaction Joris Degroote, Sebastiaan Annerel, and Jan Vierendeels

Abstract In previous work of the authors, Fourier stability analyses have been performed of Gauss–Seidel iterations between the flow solver and the structural solver in a partitioned fluid-structure interaction simulation. These analyses of the flow in an elastic tube demonstrated that only a number of Fourier modes in the error on the interface displacement are unstable. Moreover, the modes with a low wave number are most unstable and these modes can be resolved on a coarser grid. Therefore, a new class of quasi-Newton methods with more than one grid level is introduced. Numerical experiments show a significant reduction in run time.

1 Introduction Partitioned fluid-structure interaction (FSI) simulation techniques solve the flow equations and the structural equations separately. Strongly coupled partitioned techniques enforce the equilibrium of the stress and velocity (or displacement) on the fluid-structure interface in each time step. Several strongly coupled partitioned techniques are able to couple “black-box” solvers, for example the Interface Block Quasi-Newton technique with an approximation for the Jacobians from Least-Squares models (IBQN-LS) [5] and the Interface Quasi-Newton technique with an approximation for the Inverse of the Jacobian from a Least-Squares model (IQN-ILS) [2]. The coupling iterations of strongly coupled partitioned techniques can suffer from stability issues. Degroote et al. [1, 3] performed a Fourier decomposition of the error on the interface displacement during Gauss–Seidel iterations. These analyses show that only a fraction of all Fourier modes is unstable, notably the modes with a low wave number. These unstable modes can be resolved on a coarser grid. Therefore, the new Multi-Level (ML) coupling techniques presented in this work use more than one grid level, each with a different number of grid points. J. Degroote (B) Department of Flow, Heat and Combustion Mechanics, Ghent University, 9000 Ghent, Belgium e-mail: [email protected]


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Below, Multi-Level IQN-ILS (ML-IQN-ILS) is derived from IQN-ILS; Multi-Level IBQN-LS (ML-IBQN-LS) can be derived from IBQN-LS in a similar way.

2 Governing Equations A Dirichlet–Neumann decomposition of the fluid-structure interaction problem is applied. Consequently, the flow and structural solver can be represented by the functions y = F (x)

and

x = S (y),

(1)

respectively. The vector x represents the displacement of all nodes on the interface and the vector y represents the stress on all edges/faces of the interface. As the multi-level coupling algorithms use several grid levels for the flow equations and the structural equations, data has to be interpolated between different discretizations of the fluid-structure interface. Even though the discretization of the interface inside the flow solver and the structural solver depends on the grid level, all operations of the coupling algorithm are performed on a single grid – the so-called “coupling grid” (see Fig. 1) – as the interpolation is hidden inside the solvers. In this work, this coupling grid is identical to the interface discretization of the finest fluid grid. Local radial basis function interpolation is used on the interface [6]. If, for example, a stress component has to be interpolated from a fluid grid to the coupling grid, then three steps are performed for each point of the coupling grid. First, a given number of points on the fluid grid with the smallest Euclidean distance to the point of the coupling grid are selected. Then, the interpolation coefficients are calculated, using φ(z) = (1 − z)4+ (4z + 1) as basis function [6], with z the Euclidean distance divided by the radius. The plus-sign behind the first term denotes that this term is

Fig. 1 Abstract representation of coarse (light) and fine (dark) grid levels in the structural solver (left) and the flow solver (right), together with the coupling grid (centre)

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zero if 1 − z < 0 such that φ has a compact support. In this work, these first two steps are performed only once at the beginning of the simulation. Finally, the stress component is interpolated, using the interpolation coefficients from the second step.

3 Coupling Algorithm In this section, the standard IQN-ILS algorithm [2] with one grid level is first explained and subsequently extended to the ML-IQN-ILS algorithm. A prime denotes the Jacobian matrix of a function and a hat refers to an approximation. The output of a solver is indicated with a tilde. The grid level is indicated with a subscript i, with the first grid level the coarsest one and the gth grid level the finest one. The coupling iteration within time step n + 1 is denoted with a superscript k. The standard IQN-ILS coupling technique solves the FSI problem reformulated as a set of nonlinear equations in the interface’s displacement R(x) = S ◦ F (x) − x = 0

(2)

by means of quasi-Newton iterations

k −1 −rk , xk+1 = xk + R

(3)

using an approximation for the product of the inverse of the Jacobian matrix with the vector −rk . The residual is calculated as rk = R(xk ) = S ◦ F (xk ) − xk = x˜ k − xk . The coupling iterations in the time step have converged when ||rk ||2 ≤ εo with εo the convergence tolerance. The vector −rk is the difference between the desired residual, i.e. 0, and the current residual rk and it is further denoted as Δr = 0 − rk = −rk . The matrixvector product is approximated using information obtained during the previous quasi-Newton iterations. To that end, the matrices

Vk = Δrk−1 Δrk−2 . . . Δr1 Δr0

Wk = Δ˜xk−1 Δ˜xk−2 . . . Δ˜x1 Δ˜x0

(4a) (4b)

are constructed, with Δrm−1 = rm −rm−1 and Δ˜xm−1 = x˜ m −˜xm−1 (m = 1, . . . , k). These matrices contain the differences between, respectively, residual vectors and outputs of the structural solver in consecutive coupling iterations. The change of the residual vector (Δr) is decomposed as a linear combination of the differences between previous residual vectors Δr ≈ Vk ck

(5)

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with ck the coefficients of the decomposition. As the matrix Vk normally contains more rows than columns, the decomposition coefficients ck are calculated by solving a least-squares problem, using a QR-decomposition of Vk . The change of the output of the structural solver (Δ˜x) that corresponds to the change of the residual (Δr) is subsequently calculated as a linear combination of the previous changes of the output of the structural solver (Δ˜xm−1 , m = 1, . . . , k), analogous to Eq. (5), giving Δ˜x = Wk ck .

(6)

From rk = x˜ k − xk , it follows that Δr = Δ˜x − Δx, so Δx = Wk ck − Δr.

(7)

This can be interpreted as a procedure to calculate the approximation for the product of the inverse of the Jacobian matrix with the vector Δr = −rk . Algorithm 1 shows the Multi-Level IQN-ILS (ML-IQN-ILS) algorithm. Line 6–13 are the standard IQN-ILS algorithm as described above. Around the standard algorithm, an additional loop over the grid levels is added (line 3). First, the coupled solution is calculated on the coarsest grid level. Then, the solution is transferred to the next grid level on line 15–19, followed by coupling iterations on that grid level. These steps are subsequently repeated for all grid levels until the solution on the finest grid has been found. The transfer of the displacement to the following grid level provides the initial value for the coupling iterations on that grid level. The variable * (line 1) ensures that at least one coupling iteration is performed on each grid level. Because the coupling algorithm operates on a coupling grid, the difference between r and x˜ in consecutive coupling iterations is always interpolated to a fixed number of grid points, regardless of the current grid level. As a result, the modes that have been generated on a coarse grid level can be used to accelerate the coupling iterations on the finer grid levels. The same least-squares model is used for all grid levels so the number of columns in the matrices Vk and Wk increases on each grid level. Because the matrices Vk and Wk have to contain at least one column, a relaxation with a constant factor ω (line 7) is performed in the second coupling iteration of each time step. The counter k is only set to zero at the beginning of the time step (line 1) and not when the coupling iterations on a following grid level start. As a result, the relaxation step is only performed on the coarsest grid level. However, it should be noted that the difference between r and x˜ in the last coupling iteration on a certain grid level i and the first coupling iteration on the following grid level i + 1 should not be added to Vk and Wk , as these differences are biased because the terms have been calculated on two different grid levels. Line 21–23 show that synchronization is necessary at the end of the time step. Once the solution has been found on the finest grid level, all degrees of freedom on the coarser grid levels have to be corrected.

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Algorithm 1 One time step with the multi-level IQN-ILS (ML-IQN-ILS) algorithm 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23:

k=*=0 r01 = S 1 ◦ F 1 x01 − x01 = x˜ 01 − x01 for i = 1 to 7 g7do while 7rik 72 > εi,o or * = 0 do *=1 if k = 0 then xik+1 = xik + ωrik else 7 7 ck = arg minck 7Vk ck + rik 72 xik+1 = xik + Wk ck + rik end if

rik+1 = S i ◦ F i xik+1 − xik+1 = x˜ ik+1 − xik+1 k =k+1 end while if i < g then k = xik xi+1 k ri+1 = rik *=0 end if end for for i = 1 to g − 1 do start synchronizing F i and S i with F g and S g end for

4 Numerical Results To assess the performance of the multi-level algorithms, the propagation of a pressure wave in a straight flexible tube is simulated [4]. Pressure contours on the fluidstructure interface are shown in Fig. 2. Table 1 lists the number of coupling iterations per time step and per grid level, averaged over the entire simulation, and the relative duration of the simulations. The coarse grid level contains 34944+1824 degrees of freedom for the flow and the structure, respectively, while the fine grid level contains 2247168+28032 degrees of

Fig. 2 Pressure contours (in Pa) on the fluid-structure interface after 10−3 s (left), 5 × 10−3 s (centre) and 9 × 10−3 s (right)

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Table 1 Comparison between one and two grid levels for the propagation of a pressure wave in a 3D tube Algorithm Coarse iters Fine iters Duration IQN-ILS ML-IQN-ILS IBQN-LS ML-IBQN-LS

– 12.1 – 12.5

13.2 7.0 13.3 6.6

1.9 1.0 2.0 1.0

freedom. In the simulation with two grid levels, the number of coupling iterations on the fine grid is reduced by approximately 50% compared to a simulation with a fine grid only. As the cost of the coupling iterations on the coarse grid level is relatively small, the duration of the simulation also decreases by approximately 50%.

5 Conclusion Stability analyses on Gauss–Seidel coupling iterations demonstrated that the Fourier modes with a low wave number in the error on the interface displacement are most unstable. The new multi-level algorithms resolve these modes with a low wave number on a coarser grid. The numerical results show that these multi-level algorithms can reduce the duration of a partitioned fluid-structure interaction simulation, if the difference in number of degrees of freedom between the grid levels is sufficient. Acknowledgements J. Degroote gratefully acknowledges a Ph.D. fellowship of the Research Foundation–Flanders (FWO).

References 1. Degroote, J., Annerel, S., Vierendeels, J.: Stability analysis of Gauss-Seidel iterations in a partitioned simulation of fluid-structure interaction. Comput. Struct. 88, 263–270 (2010) 2. Degroote, J., Bathe, K.J., Vierendeels, J.: Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction. Comput. Struct. 87, 793–801 (2009) 3. Degroote, J., Bruggeman, P., Haelterman, R., et al.: Stability of a coupling technique for partitioned solvers in FSI applications. Comput. Struct. 86, 2224–2234 (2008) 4. Formaggia, L., Gerbeau, J.F., Nobile, F., et al.: On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels. Comput. Meth. Appl. Mech. Eng. 191, 561–582 (2001) 5. Vierendeels, J., Lanoye, L., Degroote, J., et al.: Implicit coupling of partitioned fluid-structure interaction problems with reduced order models. Comput. Struct. 85, 970–976 (2007) 6. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389–396 (1995)

Entropic Lattice Boltzmann Simulation for Unsteady Flow Around two Square Cylinders Arranged Side by Side in a Channel Takahiro Yasuda and Nobuyuki Satofuka

Abstract Entropic lattice Bolzmann model (ELBM) was developed in recent years. The method enhances stability by satisfying the second principle of the thermodynamics by imposing the monotonicity and the minimality of the H -function. But the qualitative and quantitative validities of ELBM results, especially for unsteady flow, were not confirmed sufficiently. In our previous study, we applied ELBM to a 2-dimensional channel flow past single square cylinder, and found that the validity of ELBM is guaranteed under the condition of rms value of viscosity difference ratio Δ0rms , Δ0rms ≤ 0.24. But the generality of the condition has not been confirmed. In this study, we applied ELBM to a 2-dimensional channel flow past two square cylinders arranged side by side in Reynolds number region Re = 100 and 1000 and investigated the effect of number of cylinder. As a results, it was found that as well as the in the case of single square cylinder, the condition Δ0rms ≤ 0.24 is also almost reasonable in the case of two square cylinder.

1 Introduction Entropic Lattice Boltzmann model (ELBM) is the method which overcomes the disruptive non-linear instability occurring in the standard LBGK model by satisfying the second principle of the thermodynamic by imposing the monotonicity and the minimality of the H -function. It was reported by Toshi et al. [3] that ELBM can get the stable solution for the flow in high Reynolds number regime even in the calculation with coarse mesh. But the qualitative and quantitative validity, especially for unsteady flow, were not studied sufficiently. In previous our study [2], we applied ELBM to a 2-dimensional channel flow past single square cylinder in Reynolds number range Re ≤ 1000 and it was found that the validity of ELBM is guaranteed under the condition for rms value of viscosity difference ratio Δ0rms , Δ0rms ≤ 0.24. But the generality of the condition for other computational geometries and boundary conditions has not been confirmed. In this study, we T. Yasuda (B) The University of Shiga Prefecture, Hikone, Japan e-mail: [email protected]


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applied ELBM to a 2-dimensional channel flow past two square cylinders arranged side by side in Reynolds number region Re = 100 and 1000 and investigated the effect of number of cylinder on the validity condition.

2 Numerical Method 2.1 Outline of Entropic Lattice Boltzmann Model In this study, we treat two-dimensional nine-speed (D2Q9) ELBM. ELBM is different mainly on two points from standard LBGK model. First, the equilibrium distribution function of ELBM is derived from minimization of H function under the conserving of mass and momentum. The discrete H function is given as follows, H (f) =

q−1 i=0

fi f i ln wi

(1)

where, f i is the distribution function in i direction, and q is the number of direction of speed. By calculating minimization problem of Eq. (1), the local velocity eq equilibrium distribution function in i direction f i is obtained as follows,

eq

fi

⎧ ⎤ci j ⎫ ⎡ ⎪ d ⎪ ⎨ ⎬

2u j + 1 + 3u 2j H 2 ⎦ ⎣ = wi ρ 2 − 1 + 3u j ⎪ ⎪ 1−uj ⎭ j=1 ⎩

(2)

where, ρ is the fluid density, d is the number of spatial dimension, u j is the component of macroscopic velocity in j direction. In the second point, the relaxation time of ELBM is locally adjusted in such a way that the monotonicity of the H -function is satisfied through the parameter α. The parameter α is determined by solving following non-linear equation; H ( f ) = H ( f + αΔ)

(3)

where, Δ represents the local non-equilibrium value of distribution function, f eq − f . Once the parameter α is given, the distribution function at new time step can be obtained by following lattice BGK equation; f i (x + ci , t + 1) = f i (x, t) +

α eq f (x, t) − f i (x, t) 2τ0

(4)

where, τ0 is the relaxation time in LBGK model. As described in Eq. (4), The relaxation time in ELBM, τ , becomes τ = 2τ0 /α, and, especially, in the case of α = 2, ELBM is equivalent to LBGK model.

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2.2 Numerical Conditions Figure 1 shows the definition of computational domain in present calculation. As boundary condition, parabolic velocity profile with the maximum velocity at the channel center Umax of 0.058 which corresponds to Mach number of 0.1, is imposed at the inlet boundary, half way bounce back condition is given on the channel walls and the cylinder surfaces, and at outlet boundary, the distribution functions are extrapolated from the upstream side. Five kinds of the uniform meshes, with the number of mesh points of 300 × 48 ∼ 1500 × 240 in channel length and channel width, respectively, are used in the present calculation. The definition of the Reynolds number used in this study is based on Umax , D and ν0 as follows, Re =

2τ0 − 1 Umax D ; ν0 = ν0 6

(5)

where, ν0 is the kinematic viscosity of the fluid in the case of α = 2.

8

Fig. 1 Computational domain

3 Results and Discussions 3.1 Qualitative Validity In order to validate ELBM results qualitatively, we compared vorticity distributions between ELBM and LBGK model. At Re = 100 in which LBGK models can get stable solution with the coarsest grid resolution we tested, as shown in Fig. 2, ELBM result represents that the clockwise and counter clockwise vortices are shed from upper sides and lower sides of the both cylinders, and as then the vortices are convected downstream, then the vortices gradually arrange like Karman vortex street. The flow pattern agrees well with that of LBGK model. From the results, it was found that in the case of low Re number regime at which calculation using LBGK model can get stable solution, the results of ELBM consist with those of LBGK model, and the tendency are the same as those in the case of single square cylinder.

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(a)

(b) Fig. 2 Comparison of vorticity distribution between ELBM and LBGK model (Re = 100). (a) ELBM (500 × 80), (b) LBGK model (500 × 80)

In general, increasing of Reynolds number requires higher grid resolutions in order to get stable solution. At Re = 1000, in the case of LBGK model, at least the mesh size of 1500 × 240, which is larger than 1200 × 192 of single square cylinder case, is required. Whereas in the case of ELBM, only the size of 300 × 48, as well as the single cylinder case, is required. Vortex distributions of each grid resolutions are compared in Fig. 3. It was reported that in the region Re > 300, the vortex arrangement behind the cylinder becomes irregular due to the transition to turbulence and the emergency of 3-dimensional flow structure [1]. Our ELBM results using all grid resolutions reproduce such irregular vortex shedding, however, the size of vortices in the case of 300 × 48 as shown in Fig. 2a becomes larger than that of other mesh case, therefore, it is found that the ELBM simulation using much coarser mesh leads to different flow pattern from one obtained by using finer grid. This tendency is also the same as the single cylinder case.

(a)

(b)

(c) Fig. 3 Comparison of vortisity distribution between various numbers of grid points (Re = 1000). (a) ELBM (300 × 48), (b) ELBM (600 × 96), (c) ELBM (1200 × 80)

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3.2 Quantitative Validity and Validity Condition In order to validate the results quantitatively, we investigate the time history of global enstrophy E n by the ELBM calculations with the various mesh sizes and one by LBGK calculations with the mesh of 1500 × 240, respectively, as shown in Fig. 4a. It can be seen from the figure that in the ELBM results, as grid resolution increases, the global enstrophy decreases and approaches to LBGK results with high grid resolution. It is noted that the result of E n of mesh size 300 × 48 becomes 2.5 times as large as that of mesh size 1500 × 240. This tendency is the same as in the single cylinder case as shown in previous our study [4]. 100

0.0125 ELBM(1200 × 192) LBGK(1500 × 240)

ELBM(300 × 48) ELBM(600 × 96)

ELBM(300 × 48) ELBM(600 × 96) ELBM(1200 × 192)

0.01

10–1

En

N/Ntotal

0.0075

0.005

10–2 0.0025

0

0

50

100 Time

150

200

10–3

–1

(a)

–0.5

0

Δ0

0.5

1

(b)

Fig. 4 Time history of global enstrophy E n and histogram of viscosity difference rate Δ0 at Re = 1000. (a) Time history of global enstrophy E n , (b) Histogram of viscosity difference rate Δ0

Figure 4b shows the histogram of Δ0 for various mesh sizes, where, Δ0 is the local viscosity difference rate estimated by following equation; 0 =

ν − ν0 2τ − 1 ; ν= ν0 6

(6)

and Ntotal indicates the total number of the grid points. As shown in the figure, as the grid resolution increases, the deviation of Δ0 becomes small and the results approach to one by LBGK model i.e. Δ0 = 0. This tendency is the same as in the single cylinder case [4]. In the case of single cylinder, the reliable results can be obtained under the condition Δ0rms ≤ 0.24 which corresponds to the result of mesh size of 600 × 96 or moreover. In the case of the two square cylinders, judging from the above results, the validity is obtained in the case of mesh size of 600 × 96 or moreover in which Δ0rms ≤ 0.26. Although there is a little difference, the validity condition Δ0rms ≤ 0.24 in the case of single cylinder is also reasonable in the case of

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two square cylinders. However, the strength of vortices shed from cylinder gradually becomes weak as the grid resolution increases, therefore the exact condition should be adjusted by the required accuracy of problems.

4 Conclusions In this study, we applied ELBM to a 2-dimensional channel flow past two square cylinders arranged side by side in Reynolds number of Re = 100 and Re = 1000 and investigated the effect of number of cylinder on the validity of ELBM results and the generality of validity condition by comparison with the results in the case of single square cylinder. It was found that although ELBM can enhance the stability, in much coarser grid, the solution becomes incorrect independent of the number of the cylinder. It was also found that although the exact condition should be adjusted by the required accuracy of problems, the validity condition Δ0rms ≤ 0.24 obtained in the case of single square cylinder can also be used in the case of two square cylinders.

References 1. Ansumali, S., Chikatamaria, S.S., Frouzakis, C.E., Boulouchos, K.: Entropic lattice Boltzmann simulation of the flow past square cylinder. Int. J. Mod. Phys. C 15, 435–445 (2004) 2. Breuner, M., Bernsdorf, J., Zeiser, T., Durst, F.: Accurate computations of the laminar flow past a square cylinder based on two different methods: lattice-Boltzmann and finite-volume. Int. J. Heat Fluid Flow 21, 186–196 (2000) 3. Toshi, F., Ubertini, S., Succi, S., Chen, H., Karlin I.V.: A comparison of single-time relaxation lattice Boltzmann schemes with enhanced stability. Int. J. Mod. Phys. C 17, 1375–1390 (2006) 4. Yasuda, T., Satofuka N.: Entropic lattice Boltzmann simulation for a channel flow around a square cylinder. Proceedings of the ASCHT09 2nd Asian Symposium on Computational Heat Transfer and Fluid Flow, vol. 3, pp. 152–157 (2009)

An Eulerian Approach for Fluid and Elastic Structure Interaction Problems Koji Morinishi and Tomohiro Fukui

Abstract This chapter describes a virtual flux method for simulating fluid-structure interaction problems on a fixed Cartesian mesh. The virtual flux method is one of the sharp interface Cartesian grid methods. The numerical flux across the interface is replaced with the virtual flux so that interface conditions must be satisfied there. For fluid and elastic structure interaction problems, unsteady evolution equations for the Eulerian strain tensor of elastic solid are also solved on the fixed mesh. The fluid-elastic structure interfaces are represented using the level-set function. At first validity of the numerical method is demonstrated for fluid and rigid structure interaction problems of reciprocating engines and compressors. Fluid and elastic particle interaction simulation is also carried out.

1 Introduction Nowadays fluid-structure interaction problem is one of the major subjects in CFD (Computational Fluid Dynamics). Most of numerical methods proposed for the fluid-structure interaction problem adopt the Eulerian approach in the fluid medium and the Lagrangian approach in the solid medium. For example, ALE (arbitrary Lagrangian Eulerian) method is a straightforward strategy for simulating the fluidstructure interaction problems. The ALE method may be accurate, since the fluidstructure interface is conformed to a moving boundary of the computational grid. Conforming the grid boundary to the moving interface, is generally difficult and time-consuming especially for the interface under large deformation. Cartesian grid methods are another strategy fit for simulating the fluid-structure interaction problems. The methods usually need no grid reconstruction even if large deformation of fluid-structure interface takes place. In this study, applicability of the virtual flux method [2], which is one of the sharp interface Cartesian grid methods, is demonstrated for fluid-rigid-structure interaction problems of reciprocating engines K. Morinishi (B) Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan e-mail: [email protected]


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and compressors first. For fluid and elastic structure interaction problems, unsteady evolution equations for the Eulerian strain tensor of elastic solid are also solved on the fixed mesh. The fluid-elastic structure interfaces are represented using the level-set function.

2 Virtual Flux Method The convective terms of the Navier-Stokes equations can be evaluated simply at a regular point on the Cartesian grid as:

∂Fx ∂x

=

+ − + − − Fx q˜ i−1/2 , q˜ i+1/2 , q˜ i−1/2 Fx q˜ i+1/2 Δx

i

(1)

with proper reconstructing methods, for example, − q˜ i+1/2 = q˜ i +

1 ω0 Δq˜ i+1/2 + ω1 Δq˜ i−1/2 2

(2)

where ω0 and ω1 are proper weights and Δq˜ i+1/2 is obtained with: Δq˜ i+1/2 = q˜ i+1 − q˜ i

(3)

If an immersed solid boundary is located between the points i and i + 1 as shown in Fig. 1, the numerical fluxes of Eq. (1) must be modified so that no-slip and nopenetration velocity boundary conditions are satisfied on the solid boundary.

∂Fx ∂x

=

+

+

− − Fx q˜∗ i+1/2 , q˜∗ i+1/2 − Fx q˜∗ i−1/2 , q˜ i−1/2

i

Δx

(4)

+ − + where q˜∗ i+1/2 , q˜∗ i+1/2 , and q˜∗ i−1/2 are reconstructed with considering the immersed solid boundary conditions. For example, − q˜∗ i+1/2 = q˜ i +

Fig. 1 Stencil about an interfacial point

1 ω0 Δq˜∗ i+1/2 + ω1 Δq˜ i−1/2 2

(5)

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Fig. 2 Stencil for a one-sided extrapolation

where Δq˜∗ i+1/2 is calculated with: Δq˜∗ i+1/2 = q˜∗ i+1 − q˜ i

(6)

Here q˜∗ i+1 is obtained with one-sided first or second order extrapolating operators L as shown in Fig. 2:

q˜∗ i+1 = L q˜ i−1 , q˜ i , q˜ I B

(7)

∂ q˜ q˜∗ i+1 = L q˜ i−1 , q˜ i , ∂x I B

(8)

or

where q˜ I B and ∂∂ qx˜ are the Dirichlet and Neumann boundary conditions at the IB immersed boundary, respectively. The numerical viscous fluxes must be also modified so that no-slip and nopenetration velocity boundary conditions are satisfied on the solid boundary as:

∂Sx ∂x

= i

Sx (q˜ i , q˜∗ i+1 ) − Sx (q˜ i−1 , q˜ i ) Δx

(9)

3 Application to Reciprocating Engine In this section, a fluid-structure coupling simulation is carried out to reproduce the four strokes of a reciprocating engine. Instead of modeling the complex process of fuel combustion, a proper amount of energy is added to the Navier–Stokes equation at the beginning of the expansion stroke, to retain the engine cycle. The revolution of crank shaft is described with the following equation of motion. I (θ )

1 d I (θ ) d 2θ + 2 2 dθ dt

dθ dt

2 =M

(10)

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where θ is the crank angle, M is the moment of force acting on the crank shaft, and I (θ ) is the total inertia moment of piston, connecting rod, and crank system. For the moment of force acting on the crank shaft, the moment of pressure force acting on the piston head and the moment of load acting on the crank shaft are considered. The computation is carried out for the cylinder width D of 5.8 cm and stroke H of 5.8 cm. The intake and exhaust valves are lifted up and down as a function of the crank angle. Certain amount of energy is added to the Navier–Stokes equation at the expansion stroke to retain the angular velocity of crank at about 1,800 rpm. Figure 3 shows the velocity vectors in an exhaust stroke. The gases flow out through the exhaust port. The intake mass flux and exhaust mass flux are plotted in Fig. 4. After some initial transition cycles, the engine comes to work at the almost constant rate of 1,818 rpm. The four stroke engine cycle is clearly reproduced in the simulation.

Fig. 3 Velocity vectors in an exhaust stroke

4 Application to Reciprocating Compressor In a reciprocating compressor, the intake and exhaust valves are driven by the fluid pressure force acting on them, while the piston is driven at a constant rate. Here the fluid-structure coupling simulation is carried out to reproduce the valve response and gas flows in the reciprocating compressor. The motion of intake valve and exhaust valve is assumed to be described with the following equation of motion. I

d 2θ + kθ = M dt 2

(11)


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0.04 Intake

Mass Flux [kg/s]

0.02

0.00

–0.02 Exhaust

–0.04 0.40

0.50

0.60 Time [s]

0.70

0.80

Fig. 4 Intake and exhaust mass flux

where θ is the opening angle of intake valve θi or exhaust valve θe , k the spring stiffness, and M the moment of force acting on each valve. The computation is carried out for the cylinder diameter D of 2.5 cm and stroke H of 2.5 cm. The piston is operated at the constant revolution rate of 3,000 rpm. The pressure ratio of exhaust pressure pe to intake pressure p∞ is 10. After a few transition cycles, the compressor comes to work at an almost constant flow rate. Typical velocity vectors obtained for the intake and exhaust strokes are plotted in Figs. 5 and 6, respectively. Fresh gases flow into the cylinder through the intake port (Fig. 5), and the gases are compressed and driven out through exhaust port (Fig. 6). The work of reciprocating compressor is clearly reproduced in the simulation.

5 Fluid and Elastic Structure Interaction Problems The motion of both the fluid and elastic structure may be governed by the Navier – Stokes equations in the Eulerian approach. The stress tensor of the fluid τ f and elastic solid τs can be defined as τ f = μ f [∇uf + (∇uf )T ], τs = DA,

(12) (13)

where uf is the fluid velocity, μ f is the fluid viscosity, D is the constitutive property matrix and A is the Eulerian strain tensor. The strain tensor is updated using the following evolution equations [1].

712 Fig. 5 Velocity vectors in an intake stroke

Fig. 6 Velocity vectors in an exhaust stroke



∂A ∂A 1 + us · + A(∇uf )T + (∇uf )A = [∇uf + (∇uf )T ] ∂t ∂x 2

713

(14)

The fluid-structure interfaces are represented by the level-set function φ so that the stress tensor of the Navier-Stokes equations are obtained as τ = (1 − H (φ))τ f + H (φ)τs

(15)

where H is the Heaviside function which is 1 in the solid and 0 in the fluid. Numerical results are obtained for circular particles initially immersed at rest in a Poiseuille flow and driven downstream with deformation. The results are plotted for the particles of which Young’s elasticity modulus are 10,000 and 100 in Figs. 7 and 8, respectively. The large deformation of the particle is observed for the lower

Fig. 7 Velocity vectors about a particle with an elasticity of 10,000

Fig. 8 Velocity vectors about a particle with an elasticity of 100

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stiffness case, while the initial shape of the circular particle is almost kept for the higher stiffness case. The shear strain A x x and A x y contours for the lower stiffness case are plotted in Figs. 9 and 10, respectively. The stress and strain tensors as well as velocities and pressures in both fluid and solid mediums can be simultaneously analyzed by the present approach.

Fig. 9 Shear strain A x x contours about a particle with an elasticity of 100

Fig. 10 Shear strain A x y contours about a particle with an elasticity of 100

References 1. Gao, T., Hu, H.H.: Deformation of elastic particles in viscous shear flow. J. Comput. Phys. 228, 2132–2151 (2006) 2. Tanno, I., Morinishi, K., Matsuno, K., Nishida, N.: Validation of virtual flux method for forced convection flow, JSME Int. J. 49(4), 1141–1148 (2006)

Part XXVI

DNS/RANS Computations

Dissipation Element Analysis of Inhomogenous Turbulence Philip Schaefer, Markus Gampert, Jens Henrik Goebbert, and Norbert Peters

Abstract Gradient trajectories recently received attention in the context of dissipation elements, which are space-filling regions whose theoretical description has successfully been tested via Direct Numerical Simulations (DNS) of homogenous shear flows by Wang and Peters (J. Fluid Mech. 554: 457–475, 2006; J. Fluid Mech. 608: 113–138, 2008). In the present work, the DNS of a Kolmogorov flow is evaluated via a parallelized gradient trajectory search code to verify the theory of dissipation √ elements in inhomogenous turbulence and to analyze a new k- k L model derived by Menter et al. (Turbulence, Heat and Mass Transfer, 5, 2006).

1 Introduction Gibson [1] was the first to analyze the importance of local extreme points in turbulent scalar fields and concluded that these points play a paramount role in the understanding of turbulent mixing. Recently, his findings have become relevant in the context of dissipation element analysis developed by Wang and Peters [6, 8]. Proceeding from every point in space in the direction of ascending and descending gradient direction of the underlying scalar field, one eventually ends in a local maximum, respectively minimum point. The ensemble of all points whose gradient trajectories share the same two extreme points defines a unique volume, which is called a dissipation element. As by definition one physical point only possesses a single trajectory, dissipation elements are space-filling and thus allow subdividing the field in a non-arbitrary way. Since dissipation elements arise as natural geometrical flow structures, one can expect that their understanding sheds some light on turbulence itself. Although dissipation elements are highly convoluted, Peters and Wang [6] have found that they are on average elongated with a mean diameter of the order of the Kolmogorov scale η, while the mean of their length is of the order of the Taylor scale λ. Statistically, dissipation elements can be described by two parameters, namely their linear length l and the difference of the scalar value Φ P. Schaefer (B) Institut für Technische Verbrennung, RWTH Aachen, 52056 Aachen, Germany e-mail: [email protected]


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Fig. 1 Examples of the shape of dissipation elements

Φmax

l l

Φmin

Φmin

at their ending points, cf. Fig. 1. The normalized time evolution of the probability ˜ l) ˜ has successfully been modeled [8] by the density function (pdf) of the length P( following non-dimensionalized evolution equation

˜ l, ˜ t˜) ∂ ˜ ˜ ∂ P( ˜ + a(l) ˜ l˜ = Λs P(l, t˜) v D (l) + ∂ t˜ ∂ l˜

l˜

∞

˜ l, ˜ t˜), y˜ P( y˜ , t˜)d y˜ − Λa l˜P( (1)

˜ the strain in which v D denotes the drift velocity due to molecular diffusion, a(l) rate due to the velocity difference at the ending points, Λs the so-called splitting frequency of the elements due to creation and annhilation of extreme points and Λa the so-called reattachment frequency of the elements. The theory of dissipation elements has successfully been tested against DNS data of homogenous shear flows. The analyses include various scalar fields, such as a passive scalar, the turbulent kinetic energy and the scalar dissipation. The purpose of the present work is twofold: on the one hand, inhomogenous turbulent fields are analyzed via dissipation elements, and on the other hand these results are used to calculate the constants of a modified k-ε model.

2 Dissipation Element Analysis Using a parallelized DNS code that scales up to 16.385 CPUs on an IBM BlueGene/P architecture installed at the Forschungszentrum Jülich in Germany, three different Kolmogorov flows in a 2π periodic box were simulated, cf. Table 1. Kolmogorov flows posses a sinusoidal mean velocity component in one direction of the form U = F sin(y), where F denotes the amplitude of the forced profile. Such a flow is attractive as it contains two distinct regions which are dominated by either the first or the second derivative of the mean velocity. All cases fully resolve the Kolmogorov scale, which has been identified as a necessary condition in order for the statistics of the dissipation elements to be grid independent.

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DNS case

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Table 1 Parameters of the different DNS cases 1 2

No. of grid cells Viscosity ν Amplitude F Kinetic energy k Dissipation ε Kolmogorov scale η Taylor length λ Reynolds number Reλ Resolution Δx/η

5123 5 · 10−4 0.5 0.073 0.006 0.012 0.247 110.4 0.93

10243 3 · 10−4 0.5 0.107 0.008 0.007 0.189 178.3 0.81

3 10243 2.5 · 10−4 0.5 0.115 0.010 0.006 0.170 188.5 0.97

The following analysis has been seperately conducted in the above-mentioned regions and is based on the instantanous field of the turbulent kinetic energy dissipation ε. In order to evaluate each region separately, a parallelized gradient trajectory search code has been applied, cf. [2]. While the high shear region mimics a homogenous shear flow, the low shear region allows to analyze the influence of higher derivatives of the mean velocity on the statistics. Three main quantities have been analyzed, namely the mean conditional strain ˜ where u n denotes the projection of the velocity vector ˜ =< Δu n |l > /l, rate a(l) at the extreme point in direction of the linear connection, the conditional volume weighted mean dissipation within the elements ε|l and the pdf of the element ˜ length P(l). These quantities have been compared among the different cases as well as within the different regions of one case. As has already been found for homogenous shear flows [6], it could be confirmed that the normalized length distribution is Reynoldsnumber independent in both regions and that the steady solution of Eq. (1) is in good agreement with the DNS, cf. Fig. 2. Furthermore, the mean conditional strain rate ˜ was found to turn to a constant, denoted a∞ , for large element lengths and to a(l) be proportional to the mean shear rate imposed in the high shear region, cf. Fig. 3. In addition, the mean conditional dissipation rate ε|l within elements can be approximated by ε∗l˜−n , cf. Fig. 4, where the exponent n was found to be the same in both regions of the flow but turned out to be Reynolds-number dependant. Since dissipation elements are space-filling, one can reconstruct the statistics of the entire field based on the conditional statistics within elements. Knowing the conditional mean dissipation within elements as well as the length distribution of the elements, an equation for the time evolution of the entire mean dissipation field can be derived by multiplying Eq. (1) with ε∗l˜1−n , which corresponds to taking the 1 − nth moment of the length distribution. This procedure yields

∂ε = a∞ ε∗ Isplit − Iattach + Istrain + Idri f t , ∂t

(2)

in which the time is dimensionalized again and the terms on the right hand side correspond to the different integrals.

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1

100

10−2

0.8 ˜ P˜ (l)

10−4

0.6

10−6

10−8

0.4 10−10

1

2

3

4

5

model (De = 1.5)

0.2

0

1

2

3

˜l

6

7

high shear (case 1)

low shear (case 2)

low shear (case 1)

high shear (case 3)

high shear (case 2)

low shear (case 3)

4

5

6

7

8

9

10

l˜

˜ of the element length for the different DNS cases Fig. 2 Pdf P(l) 1 a∞ = 0.68

a (˜l ) 0.5

a∞ = 0.40 0

high shear (case 1) low shear (case 1) high shear (case 2) low shear (case 2) high shear (case 3) low shear (case 3)

−0.5

−1 0

1

2

3

˜l

4

5

6

Fig. 3 Normalized conditional strain rate a|l

Based on a recent work by √ Menter et al. [5], a new model equation, to be called Menter – Egorov model, for k L has been introduced, where k is the kinetic energy and L an integral length scale. Its derivation based on Rotta’s [7] approach is appealing, as it naturally √ introduces higher derivates of the mean velocity into the production term. The k L equation can easily be transformed into an equation for the mean dissipation ε yielding

Dissipation Element Analysis of Inhomogenous Turbulence

ε⏐| >

721 high shear (case 1) low shear (case 1) high shear (case 2) low shear (case 2) high shear (case 3) low shear (case 3)

n = 0.42

10−2 n = 0.38

10−1

100

˜l

Fig. 4 Conditional mean dissipation ε|l

∂(ρε) ∂(ρU j ε) ∂ = + ∂t ∂x j ∂x j

ρνt ∂ε σ ∂x j

ε ε2 + cε1 ρ Pk − cε2 ρ k k 2

∂ ε 2 , ∂x j k

(3)

cε1 = 1.2, cε2 = 1.825, cε3 = 0.0212, cε4 = 2.0, cμ = 0.09, σ = 2/3.

(4)

k4 + cε3 ρ 2 ε

∂ 2 Ui ∂ x 2j

ρcμ k 4 − cε4 σ ε2

with the model constants

As can be seen, compared to the standard model [3], this model contains an additional production term proportional to the second derivative of the mean velocity field and proposes slightly different values for the model constants than [4]. Another additional term can be identified as a cross-diffusion term, but is of minor importance for the current case. Due to the different regions of its mean velocity profile, a Kolmogrov flow naturally provides a good testing ground for such a model. Evaluating Eq. (2) in the two different regions of the Kolmogorov flow and conducting a term by term comparison with Eq. (3) yields a linear set of equations Table 2 Coefficients of the ε-equation obtained via dissipation elements DNS case 1 2 3 Reλ cε1 cε2 cε3

110.4 0.79 0.94 2.0 ·10−3

178.3 1.24 1.74 2.9 ·10−3

188.5 1.43 1.83 3.2 ·10−3

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for the model constants cε1 , cε2 and cε3 , which turn out to be Reynolds-number dependant, cf. Table 2, but asymptotically tend to the values proposed by [5] for the Menter–Egorov model, cf. Eq. (4).

3 Conclusion Based on a previous work on homogenous shear turbulence [8], a dissipation element analysis has been conducted separately in two regions of the Kolmogorov flow field; one is dominated by the first derivative of the mean velocity, thereby locally resembling a homogenous shear flow, whilst the other is dominated by the second derivative. The instantanous dissipation field was used as the underlying scalar. Dissipation elements have been identified within the two regions and were used to subdivide the field into space-filling regions. It could be shown that, as has already been found for homogenous turbulence, the normalised length distribution of dissipation elements is Reynolds-number independent in both regions of the flow. The conditional mean strain rate due to velocity differences at the ending points was found to scale linearly with the separation length and is of the order of the mean shear of the high shear region for all Reynolds numbers. Multiplying and integrating the pdf transport equation of the linear length with the volume averaged dissipation within the elements yields an evolution equation for the overall dissipation ε. A term-by-term comparison with the Menter–Egorov model allows the calculation of the modelling constants. While the values for cε1 and cε2 of the highest Reynolds number case turn out to be close to those proposed in [5], the value for cε3 is largely underpredicted. Acknowledgements This work has been funded by the Deutsche Forschungsgemeinschaft under grant Pe 241/38-1. Furthermore, the project has been continuously sponsored by NIC at the Research Centre Jülich. The authors greatly appreciate the necessary support.

References 1. Gibson, C.H.: Fine structure of scalar fields mixed by turbulence. I. Zero gradient points and minimal gradient surfaces. Phys. Fluids 11, 2305–2315 (1968) 2. Schaefer, P., Gampert, M., Goebbert, J.H., Wang, L., Peters, N.: Testing of model equations for the mean dissipation using Kolmogorov flows. Flow, Turbulence and Combustion 85, 225–243 (2010) 3. Jones, W.P., Launder, B.E.: The Prediction of laminarization with a two-equation model of turbulence. Int. J. Heat Mass Transf. 15, 301–314 (1972) 4. Launder, B.E., Sharma, B.I.: Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transf. 1, 131–1318 (1974) √ 5. Menter, F.R., Egorov, Y., Rusch, D.: Steady and unsteady flow modeling using the k- kL model. In: Turbulence, Heat and Mass Transfer 5, 403–406 (2006) 6. Peters, N., Wang, L.: The length scale distribution function of the distance between extremal points in passive scalar turbulence. J. Fluid Mech. 554, 457–475 (2006) 7. Rotta, J.C.: Turbulente Strömungen. Teubner, Stuttgart (1972) 8. Wang, L., Peters, N.: Length scale distribution functions and conditional means for various fields in turbulence. J. Fluid Mech. 608, 113–138 (2008)

A Matrix-Free Viscous Linearization Procedure for Implicit Compressible Flow Solvers Nikhil Vijay Shende and N. Balakrishnan

Abstract In this work, a matrix (inversion) free exact viscous linearization procedure for implicit compressible flow solvers in unstructured mesh framework is developed. Novel feature of this procedure is to solve RANS equations for a finite volume in a coordinate system aligned to the Principal axes of viscous stress tensor. For RAE aerofoil, it is demonstrated that the proposed procedure is more robust compared to a conventional matrix-free procedure. Though, in its present form, viscous linearization procedure is expensive, but it definitely adds to the robustness of flow solver. An intelligence built into the flow solver to selectively employ the Principal axes transformation is expected to render the code both robustness and efficiency.

1 Introduction Robustness, efficiency and accuracy are the three important features of an industry standard CFD solver. While the efficiency of a CFD solver is exhibited in its rapid convergence to steady state, the robustness ensures trouble free convergence in an automated cycle over a range of geometric complexities and operating conditions. It is well known that both these features are related and the convergence acceleration tools like the implicit procedure are critical to achieve both these desired features. Interestingly, the accuracy, the third of the above features, is often attributed only to the discretization schemes, missing out the important role played by the implicit solvers in ensuring convergence to steady state in the dual loop of an unsteady solver, which is critical to achieve the required time accuracy. Therefore, in our view, a robust implicit procedure is critical to achieve all the three desired features both in steady and unsteady CFD solvers. In this chapter we present an effort in this direction. It is well known in CFD literature, an exact linearization is essential to achieving a quadratic convergence. In reality, this may be difficult, because of the algebraic N.V. Shende (B) Computational Aerodynamics Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India e-mail: [email protected]


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complexity of the exact linearization or higher computational cost associated with the block matrix inversion or loss in diagonal dominance which is critical to an iterative solver. This has resulted in the search for approximate or inconsistent linearization tools which are both robust and economical. Therefore, this area is well researched and matrix-free implicit solvers [3] constitute one of the significant developments in this area. Particularly in this context, it is worthwhile to recall the research efforts which culminated in the Lower–Upper symmetric Gauss–Seidel (LU–SGS) procedure [2, 7] for iterative inversion of the implicit matrix. The split flux Jacobian proposed in this work [7] was later exploited within a finite volume frame work for building a matrix-free implicit procedure [3]. The importance of this procedure stems from the fact that the dense block matrices are replaced by diagonal matrices, thus obviating any matrix inversion and thereby the resulting procedure is computationally cheap. This approximate linearization procedure, interestingly works well with most of the upwind schemes for explicit residual computation. Further performance enhancement of this procedure by the way of employing several Point Jacobi (PJ) or symmetric Gauss–Siedel (SGS) iterations has also been reported [5]. The damping property of such procedures, particularly when a multiplication factor to the spectral radius of the flux Jacobian is introduced while building the split flux Jacobians, has also been studied [4]. It worthwhile to note that this procedure is limited to the treatment of inviscid fluxes and the viscous flux linearization is achieved in rather a simplistic way by adding a viscous velocity

μ to the spectral radius of the flux Jacobian. In spite of the simplicity of the scale ρl aforementioned procedure, it is indeed very effective for a wide range of problems. At the same time, there are occasions, particularly involving high Reynolds number flows, invariably associated with high levels of grid induced stiffness, where this methodology is not adequate to render the required level of robustness to the code. This inspires us to search for a better linearization which not only results in a robust convergence to steady state but also retains the simplicity and the cost effectiveness of the matrix-free implicit procedure. In this work we explore the possibility of retaining the inconsistent linearization for the inviscid fluxes resulting in a matrixfree procedure, while an exact linearization is attempted for the viscous fluxes. The exact linearization of the viscous fluxes is made cost effective by the way of employing a Principal axis transformation which diagonalizes the viscous stress tensor. It is demonstrated that the associated block implicit matrix is effectively lower triangular, thus the matrix inversion is obviated. It should be remarked that we still compute the components of a lower triangular matrix and the method is matrix free in the sense that it does not require any matrix inversion. The theme of the present work is to study the usefulness of such a procedure and its cost effectiveness.

2 Methodology The distinctive feature of the present procedure is effecting Principal axis transformation of the local coordinate system and solving RANS equations on the transformed coordinates. By this: (1) All the shear stress terms in the stress tensor vanish

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resulting in a relatively easy linearization procedure. (2) It is possible to cast the viscous flux Jacobian as a lower triangular matrix, thereby avoiding the matrix inversion for the block elements of implicit matrix and retaining the matrix free operation. For any given cell i in the computational domain, the angle θ made by one of the Principal axis corresponding to viscous stress tensor with horizontal is given by: θ =

1 −1 2 tan

∂u ∂v ∂y + ∂x ∂u ∂v ∂x − ∂y

where

∂u ∂u ∂v ∂v ∂x , ∂y , ∂x , ∂y

are the gradients of Cartesian veloc-

ity components. The vector–matrix equation for cell i using backward Euler time integration procedure in Principal axes can be written as

Ωi n+1 n+1 Δt Wi + F⊥J ΔS J = 0, with Δt (·) = (·)n+1 − (·)n . + F⊥J Δt

(1)

J

In this equation, Ωi and Wi are volume and vector of conserved variables corresponding to cell i respectively, ΔS J is the area of face J belonging to cell i, Δt is the time step, F⊥J is the convective flux and F⊥J is the viscous flux. The convective flux can be calculated using a suitable upwind procedure and the viscous flux can be calculated using a suitable centred procedure. In order to solve this equation, it is necessary to linearize inviscid and viscous fluxes. The linearization of inviscid flux leads to following equation: +,n −,n n+1 n F⊥J = F⊥J + Ai,⊥J Δt Wi + A j,⊥J Δt W j

(2)

± are the inviscid split normal flux Jacobians. In the present work, we where Ai,⊥J ± = use the inviscid split flux Jacobians given by Yoon and Jameson [7] as A⊥J 1 ± ρ I . The viscous flux normal to interface J of cell i, in Principal A ⊥J A⊥J 2 axes (ζ − η), is given as

T F⊥J = 0, −τζ ζ n ζ , −τηη n η , −u ζ τζ ζ n ζ − u η τηη n η + qζ n ζ + qη n η . (3) The linearization of second component of viscous flux vector can be given as follows: (2),n+1 F⊥J

(2),n

= F⊥J

+

2 ∂u η 4 ∂u ζ μ − μ = −τζ ζ n ζ = − 3 ∂ζ 3 ∂η (2) ∂F⊥J ∂ T3

∂ T3 ∂Δt W ∂ T3 Δt W + ∂W ∂Wζ ∂ζ

n+1

+ J

nζJ J (2) ∂F⊥J ∂ T4

2 4 μT n+1 − μT4n+1 =− 3 3 3 ∂ T4 ∂Δt W ∂ T4 Δt W + ∂W ∂Wη ∂η

(4) J

Evaluating gradients appearing in above equation using Green–Gauss theorem based diamond path reconstruction for face J formed by the nodes N1 and N2 shared by cells i and j we have ∂Δ∂ζt W = ai Δt Wi + aj Δt Wj + a N1 Δt W N1 + a N2 Δt W N2 and ∂Δt W ∂η

= bi Δt Wi + bj Δt Wj + b N1 Δt W N1 + b N2 Δt W N2 , with a and b being the

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geometric weights obtained by rearranging the terms in Green–Gauss procedure. Substituting the gradients formulae in Eq. (4) and rearranging, we have (2),n+1

F⊥J

(2),n

= F⊥J

+ Pi Δt Wi + P j Δt W j + P N1 Δt W N1 + P N2 Δt W N2 .

(5)

In above equation, P stands for the row vector. Extending aforesaid procedure for other components of the viscous flux vector, the linearized equation can be given in compact form as n+1 n F⊥J = F⊥J + Mi Δt Wi + M j Δt W j + M N1 Δt W N1 + M N2 Δt W N2

(6)

with M being the viscous coefficients matrix. Substituting Eqs. (2) and (6) in Eq. (1), we get ⎡

⎤ 1 + I 1 1 − ⎣ + Ai,⊥J ΔS J + Mi ΔS J ⎦ Δt Wi + A j,⊥J ΔS J Δt W j Δt Ωi Ωi Ωi J

J

J

1 1 + M j ΔS J Δt W j + M N1 Δt W N1 + M N2 Δt W N2 J ΔS J = Rin Ωi Ωi J

(7)

J

Using vector algebra, evaluation of the positive split flux Jacobian using cell ρA⊥J + averaged state Wi leads to J Ai,⊥J ΔS J = J 2 I ΔS J . In addition, − using homogeneity property of inviscid fluxes, = J A j,⊥J Δt W j ΔS J 1 J Δt F j,⊥J − ρA⊥J Δt W j ΔS J . Also, matrix Mi in Eq. (7) is predominantly 2 a lower triangular matrix with non-zero entry only at its second row and third col(l) (u) (u) umn. By splitting Mi = Mi +Mi and allowing Mi to lag by one subiteration, the matrix inversion is completely eliminated. Hence, the equation for matrix inversion free procedure reads as follows 4 1 (l) I 1 ρAi,⊥J ΔS J + Mi ΔS J Δt Wi + + Δt 2Ωi Ωi J J

1 Δt F j,⊥J − ρA j,⊥J Δt W j ΔS J + 2Ωi J 1 (u) 1 Mi ΔS J Δt Wi + M j ΔS J Δt W j + Ωi Ωi J J 1 M N1 Δt W N1 + M N2 Δt W N2 J ΔS J = Rin Ωi

3

(8)

J

This equation is solved using symmetric Gauss Seidel (SGS) relaxation procedure [4].

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3 Results and Discussion The efficacy of the proposed viscous linearization procedure is demonstrated using transonic turbulent flow past RAE aerofoil. The eddy viscosity is computed using Spalart–Allmaras turbulence model [6]. The free stream Mach number is 0.729, angle of attack is 2.79o and free stream Reynolds number is 6.5×106 . Two grids are employed for the present study. Grid 1 is an unstructured grid having quadrilateral elements with 28,093 points, 27,793 cells and 55,886 faces. Grid 2 is a structured C– type grid with 21,920 points, 21,608 cells and 43,528 faces. Figure 1 depicts grids 1 and 2 used for the present computations. For all the computations, CFL number is progressively increased as million times iteration number and local time stepping is employed. The solution is declared to be converged to steady state when the relative residue of density falls by ten decades. At this stage, the relative residue of modified turbulence viscosity (ν˜ ) converges to about eight and half decades. Different procedures employed in the present study are: (1) conventional procedure–Standard matrix-free implicit procedure [3], (2) VL procedure–Present procedure where viscous linearization is invoked from first iteration and Principal axes are computed at every iteration during solution evolution, (3) VL variant-1–where viscous linearization is invoked only after the density residue is converged by 5 decades and Principal axes are computed once in 50 iterations. Table 1 gives the comparison of Cl and Cd obtained for grid 1 using conventional, VL and VL variant-1 procedures with experimental results [1] and total time required for convergence. From this table, it can be seen that the VL procedure is about 5 times expensive compared to conventional procedure while VL variant-1 is about 2.5 times expensive compared to conventional procedure. For grid 2, it is observed that conventional procedure breaks down after 6 decades of residue fall in density. On the contrary, both VL and VL variant-1 procedures on grid 2 converge to steady state. Table 1 also gives Cl and Cd obtained for grid 2 using aforesaid procedure and total time required for convergence. From this table, it can be seen that VL variant-1 is about 1.75 times faster compared to VL procedure. Finally, this case clearly demonstrates the robustness of VL procedures over conventional procedure.

Fig. 1 RAE aerofoil: Grid 1 (left) and Grid 2 (right)

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N.V. Shende and N. Balakrishnan Table 1 Comparison of Cl , Cd and total time for convergence Grid Lift coefficient, Cl Drag coefficient, C d Time in minutes

Conventional VL VL variant-1

Grid-1 Grid-1 Grid-1

0.8011 0.8011 0.8011

0.0176 0.0176 0.0176

83 411 204

VL VL variant-1 Experiments

Grid-2 Grid-2 ...

0.7873 0.7873 0.8030

0.0167 0.0167 0.0168

139 81 ...

4 Conclusion In this work, a matrix (inversion) free exact viscous linearization procedure for implicit compressible flow solvers in unstructured mesh framework is developed. Novel feature of this procedure is to solve RANS equations for a finite volume in a coordinate system aligned to the Principal axes of viscous stress tensor. For RAE aerofoil, it is demonstrated that the proposed procedure is more robust compared to a conventional matrix-free procedure. Though, in its present form, viscous linearization procedure is expensive, but it definitely adds to the robustness of flow solver. An intelligence built into the flow solver to selectively employ the Principal axes transformation is expected to render the code both robustness and efficiency.

References 1. Cook, P.H.: Report Number AR-138. AGARD (1979) 2. Jameson, A., Turkel, E.: Implicit schemes and LU decomposition. Math. Comput. 37(156), 385–397 (1981) 3. Luo, H., Baum, J.D., Lohner, R.: A fast, matrix–free implicit method for compressible flows on unstructured grids. J. Comput. Phys. 146, 664–690 (1998) 4. Naik, C.B.: A Novel Linearization procedure for Viscous Fluxes and Stability of Implicit Schemes. M.E. Thesis, Aerospace Engineering, Indian Institute of Science, Bangalore (2006) 5. Nikhil, S., Balakrishnan, N.: New migratory memory algorithm for implicit finite volume solvers. AIAA J. 42(9), 1863–1870 (2004) 6. Spalart, P.R., Allmaras, S.R.: A one-equation turbulence model for aerodynamic flows. In: AIAA Paper 92-0439 (1992) 7. Yoon, S., Jameson, A.: Lower–Upper symmetric Gauss–Seidel method for the Euler and Navier–Stokes equations. AIAA J. 26(9), 1025–1026 (1988)

DNS of Shock/Boundary Layer Interaction Flow in a Supersonic Compression Ramp Xin-Liang Li, De-Xun Fu, Yan-Wen Ma, and Xian Liang

Abstract A direct numerical simulation of the shock/turbulent boundary layer interaction flow in a supersonic 24-degree compression ramp is conducted with the free stream Mach number 2.9. The blow-and-suction disturbance in the upstream wall boundary is used to trigger the transition. Both the mean wall pressure and the velocity profiles agree with those of the experimental data, which validates the simulation. The turbulent kinetic energy budget in the separation region is analyzed. Results show that the turbulent production term increases fast in the separation region, while the turbulent dissipation term reaches its peak in the near-wall region. The turbulent transport term contributes to the balance of the turbulent conduction and turbulent dissipation. Based on the analysis of instantaneous pressure in the downstream region of the mean shock and that in the separation bubble, the authors suggest that the low frequency oscillation of the shock is not caused by the upstream turbulent disturbance, but rather the instability of separation bubble.

1 Introduction During more than half a century study, large amount of experimental and numerical study are preformed on shock/turbulent boundary-layer interaction (STBLI), and the supersonic compression ramp flow is a typical model of STBLI [3]. Adams et al. [1] performed the first DNS of supersonic compression ramp flow. However, limited by the computing power, the Reynolds number of DNS cannot be as high as that of the experiment, and there is no direct comparison with the experiment result in the DNS used by Adam. Wu and Martin [7] performed a new DNS of supersonic compression ramp flow and the turbulence statistics agree with Bookey et al.’s experiment [2]. To minimize the computation cost, Wu and Martin used the recycler technique to produce the inlet fully developed boundary-layer turbulence. This method avoided the long computational domain for simulating the transition. However, the inlet turbulence provided by the recycler technique is not as reliable X.-L. Li (B) LHD, Institute of Mechanics, CAS, Beijing 100190, China e-mail: [email protected]


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as that provided by transition simulation. This chapter has made a direct numerical simulation (DNS) of shock/turbulent boundary layer interaction (STBLI) flow in a 24-degree compression ramp with free stream Mach number 2.9. The flow parameters chosen are close to Bookey et al.’s experiment [2]. Different from Wu and Martin’s simulation [7, 8], the wall blow-and-suction disturbance in the upwind wall boundary is used to trigger the transition. Based on the analysis of instantaneous pressure in the downstream region of the mean shock and that in the separation bubble, the authors suggest that the low frequency oscillation of the shock is not caused by the upstream turbulent disturbance, but rather the instability of separation bubble. The turbulent kinetic energy budget in the separation region is also analyzed in this chapter.

2 DNS Setup As shown in Fig. 1, the computing model is supersonic flow over a 24-degree ramp, and the computational domain is also shown in this figure as the dashed line. The computational domain is 0 ≤ z ≤ 14 mm in the spanwise direction. In the current simulation, we first simulate the two-dimensional laminar flow over a flat-plate with the leading edge, then, use the profiles of density, velocities and temperature at the 200 mm downstream leading edge are used as the inlet boundary conditions of the three-dimensional simulation. To trigger the transition, we impose the blow-andsuction perturbation on the wall at −305 mm ≤ x ≤ −285 mm. The Amplitude of the perturbation is set as A = 0.1. Table 1 shows the free-stream conditions and the condition at the location x = −30 mm, which is in the upstream of the separation bubble, where θ, δ and C f denote the momentum thickness, nominal thickness (99%) and skin friction coefficient at x = −30 mm. The parameters of Bookey et al.’s experiment [2] is also listed in this table.

Fig. 1 Schematic diagram of DNS setup Table 1 Flow parameters Free-stream and wall Ma∞ The current DNS Bookey et al. [2]

2.9 2.9

Re∞ /mm 5,581.4 5,581.4

T∞ /K 108.1 108.1

Tw /K 307 307

x = −30 mm

Reθ

Cf

θ/mm

δ/mm

2,344 2,400

2.57 × 10−3

0.42 0.43

6.5 6.7

2.25 × 10−3

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The Navier–Stokes equations in the curvilinear-coordinate are used in the current DNS study. Steger–Warming splitting is used for the inviscid terms and then solved by using WENO-SYMBO method of Wu and Martin [7, 8]. Viscous terms are discretized by using the eighth order central scheme, and the third order TVD-type Runge–Kutta method is used for time-advance. Mesh of the current DNS is 2160 (streamwise) ×140 (wall-normal) ×160 (spanwise). And the mesh is concentrated in the corner region (−35 mm ≤ x ≤ 35 mm). The mesh span in the wall unit (measured at x = −30 mm) is x + ≈ 4.1, yw+ ≈ 0.5 and z + ≈ 4.8, which is much smaller than the DNS of non-separated flat-plate boundary layer.

3 Data Validation Figure 2 shows the mean wall pressure p¯ w / p∞ as a function of x/δ, where δ is the nominal boundary layer thickness at x = −30 mm. This figure shows that the mean pressure grows rapidly in the region x/δ ≥ −3.5, and then forms a platform region. The circles in Fig. 2 denote the experimental data of Bookey et al. [2], and the error bar is set at 5%, and this figure shows that the current DNS result is agrees well with the experimental data. Figure 3 shows the mean velocity profile at x = −20 mm, which is agrees the experimental data well. Figures 2, 3 validate the current DNS. 5

Exp. DNS

pw/p∞

4

3

2

1 –10

–5

0

5

x/δ

Fig. 2 Distribution of the mean wall pressure

4 Flow Visualization and Turbulent Kinetic Energy Budget Figure 4 shows the two-dimensional distribution of the instantaneous temperature in the spanwise middle section (z = 7 mm). Regions of laminar flow, transition, fully developed turbulence and separation are shown clearly in this figure. Figure 5 shows

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1

U/U∞

0.8 0.6

Exp. DNS

0.4 0.2 x = –20 mm 0 0

0.5

1 y/δ

Fig. 3 Mean velocity profile at x = −20 mm

Fig. 4 Distribution of the instantaneous temperature in the middle section: z = 7 mm

Fig. 5 Instantaneous numerical schlieren plot at t = 2,100 and t = 2,505

the instantaneous numerical schlieren plots [7] at t = 2,100 and t = 2,505, and this figure shows that the deformation of the main shock and the shocklets extends out from the boundary layer. Figure 6 shows the two-dimensional distribution of each terms in turbulent kinetic budget [6], which contains the production term (P), dissipation term (ε), turbulent transport term (T ) and pressure-dilatation term (Pdiv). This figure shows

Fig. 6 Distribution of the production term (P), the dissipation term (ε), the turbulent transport (T ) and the pressure-dilatation term (Pdiv) in the turbulent kinetic energy equation

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that the production term is very strong in the downstream region of the separation point and in the region around the main shock. This is because the mean shear is very strong in these regions. The turbulent dissipation ε is very strong both in the separation region and the downstream near-wall region. Turbulent transport term balances the production and dissipation, and the turbulent kinetic energy is transported to the near wall region and then dissipated. This figure also shows that the pressuredilatation term (Pdiv) is significant in the region around the main shock, while this term is ignorable in other regions. This indicates that the intrinsic compressibility effect is not significant in the region which does not contain the main shock.

5 Preliminary Investigation of the Mechanism for Shock Oscillation The oscillation of the separation shock is an important feature of the shock/turbulent boundary layer interaction (STBLI). There are two different characteristic frequencies of the shock oscillation, and the time scale of the high-frequency oscillation is O(δ/U∞ ), while the time scale of the low-frequency one is O(10δ/U∞ − 100δ/U∞ ) [4]. Ganapathisubramani et al. [5] have found that there are clusters of coherent structures (so-called “super-structures”) in the turbulent boundary layer and deemed that the clusters of coherent structures play an important role in the low-frequency oscillation of the shock. Figure 7 shows the visualization of coherent structures by using the iso-surface of the second invariant of velocity gradient tensor Q. This figure shows that coherent structures are randomly arranged, and the package of coherent structures are not clear. The authors tend to believe that the package (or cluster) of the coherent structures are not the reason for the low-frequency oscillation of the shock. ¯ p∞ in Figure 8 shows the time history of pressure disturbance p = ( p − p)/ the point (x, y) = (−9, 5.9), which is located in the downstream region of the shock, and the fluctuation of p denotes the oscillation of the shock. The dash and the solid lines in Fig. 8 denote the instantaneous and filtered values of p , respectively. This figure shows that low-frequency oscillation has the time scale of

Fig. 7 Visulazation of the coherent structures (isosurface of Q)


735

Filtered pressure Instantaneous pressure

1

p’

0.5

0

–0.5

–1

x = –9, y = 5.9 2600

2800

3000

3200

3400

3600

t

Fig. 8 Time history of the pressure disturbance at the location (x, y) = (−9, 5.9)

100–400 non-dimensional time units, which is approximately 15–60 times of δ/U∞ . Figure 9 shows the pressure disturbance p at the wall at x = −12 mm, which is located in the separation bubble, and it also shows the oscillation with two different frequencies. The low frequency oscillation denotes the oscillation of the separation bubble, and the time scale is close to that of the shock oscillation. So, the authors conjecture that the low-frequency oscillation of the shock is associated with the oscillation of the separation bubble, and less related to the upstream coherent structures. To further validate this speculation, we perform a laminar simulation of the

Filtered Pressure Instantaneous pressure

0.8 0.6 0.4

p'

0.2 0 –0.2 –0.4 –0.6 x = –12mm, y = 0 –0.8 2600

2800

3000

3200

3400

t

Fig. 9 Time history of the pressure disturbance at the wall of x = −12 mm

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compression ramp flow. The laminar simulation is two-dimensional with the same free-stream conditions as the current DNS, and the geometry is also the same as the current DNS (in the x–y section). Different from the three-dimensional simulation, no perturbation is used in the two-dimensional simulation and the flow remains laminar. Figure 10 shows the instantaneous pressure in the wall at x = −4 mm, and this figure shows that the oscillation of wall pressure has two different time scales. The low frequency oscillation has the time scale of 400 non-dimensional time units, which is close to the time scale of low-frequency oscillation in the turbulent case. In the laminar simulation, the flow in the upstream region of the separation bubble is “clean”, and there are no coherent structures or super-structures. However, the low-frequency oscillation of the separation bubble still exists. This indicates that the low-frequency oscillation is not related to the upstream disturbance. So, the authors suggest that the low frequency oscillation of the shock has little relation with the upstream turbulent disturbance, and the instability of separation bubble is the suggested reason. The mechanism of low-frequency oscillation will be further investigated in the follow-up study. 1.6 low - freqency

Pw/P∞

1.4

1.2

1

0.8

x = –4mm 2200

2400

2600

2800

3000

t Fig. 10 Time history of the pressure disturbance at the wall of x = −4 mm (laminar simulation)

6 Conclusion The direct numerical simulation of shock/turbulent boundary layer interaction flow in a supersonic compression ramp is conducted with free stream Mach number 2.9 and 24-degree ramp-angle. The upstream wall blow-and-suction perturbation is used to trigger the transition. Both the mean wall pressure and the velocity profiles agree with those of experimental data, which validates the simulation. The turbulent kinetic energy budget in the separation region is analyzed, and the mechanism of the main shock’s low-frequency oscillation is also studied.


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The authors suggest that the low frequency oscillation of the shock has little relation with the upstream turbulent disturbance, which results from the instability of separation bubble. The turbulent production term increases fast in the separation bubble, while the turbulent dissipation term reaches its peak in the near-wall region. The turbulent transport term contributes to the balance of the turbulent conduction and turbulent dissipation.

References 1. Adams, N.A.: Direct simulation of the turbulent boundary layer along a compression ramp at M = 3 and Re = 1685. J. Fluid Mech. 420, 47–83 (2000) 2. Bookey, P.B., Wyckham, C., Smits, A.J., Martin, M.P.: New Experimental Data of STBLI at DNS/LES Accessible Reynolds Numbers. AIAA Paper 2005-309 (2005) 3. Dolling, D.S.: Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA J. 39(8), 1517–1531 (2001) 4. Dolling, D.S., Or, C.T.: Unsteadiness of the shock wave structure in attached and separated compression ramp flows. Exp. Fluids 3, 24–32 (1985) 5. Ganapathisubramani, B., Clemens, N.T., Dolling, D.S.: Low-frequency dynamics of shockinduced separation in a compression ramp interaction. J. Fluid Mech. 636, 397–425 (2009) 6. Pirozzoli, S., Grasso, F.: Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at M = 2.25. Phys. Fluids 18(6), 065113 (2006) 7. Wu, M., Martin, M.P.: Direct numerical simulation of shockwave and turbulent boundary layer interaction induced by a compression Ramp. AIAA J. 45, 879–889 (2007) 8. Wu, M., Martin, M.P.: Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data. J. Fluid Mech. 594, 71–83 (2008)

An Efficient Generator of Synthetic Turbulence at RANS–LES Interface in Embedded LES of Wall-Bounded and Free Shear Flows Dmitry Adamian and Andrey Travin

Abstract A new simple method is presented for generating velocity fluctuations at the inflow of LES domain in Embedded LES. The method employs only RANS turbulence statistics and is shown to be rather accurate in both canonical shear flows (plane channel, zero pressure gradient boundary layer, and plane mixing layer) and in a wall-mounted hump flow with pressure-induced separation and reattachment.

1 Introduction Hybrid RANS–LES approaches to turbulence representation are now considered as the only currently manageable alternative to the pure RANS of complex turbulent flows at high Reynolds numbers. One of the most flexible approaches of this type is the so-called Embedded LES (ELES), which assumes using LES only in a restricted arbitrary specified flow region(s) where pure RANS is incapable or turbulent content of the flow is for some reason essential, whereas the rest of the flow is treated with RANS. A key prerequisite of these approaches in the case when LES region is located downstream of RANS area is a robust way to produce realistic turbulent content at the RANS–LES interface. A number of methods aimed at resolving this challenging problem have been proposed, including the use of external databases from LES or DNS of simple flows (e.g., the developed channel flow), different recycling/rescaling procedures, and “synthetic turbulence” generators. All these methods have their pros and cons. For instance, the recycling methods are capable of creating a natural inflow turbulence but are applicable only in the nearly equilibrium flow regions. The synthetic methods are, in principle, more flexible. However some of them require too detailed knowledge of turbulent statistics (length-scales, timeand space-correlation functions, etc.) which could not be provided by RANS models used upstream of the RANS–LES interface while using the other ones results in realistic turbulence structures being established too slow, thus causing significant degradation of the whole solution. D. Adamian (B) St. Petersburg State Polytechnic University, St. Petersburg 195257, Russia e-mail: [email protected]


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In the present work a new simple ad hoc algorithm is proposed for generation of “synthetic turbulence” (velocity fluctuations) at the inflow of LES domain which employs only the turbulent quantities involved in the conventional two-equation RANS models and, at the same time, ensures fairly rapid transition to physically realistic turbulence. The algorithm employs ideas of Bechara et al. [1] who proposed a harmonic generator of turbulence for stochastic noise modeling and some elements of other available turbulence generators. However, unlike these methods, it is capable of plausible representation of anisotropy of the vortical structures, which is an essential feature of the near-wall turbulence.

2 Formulation Let U (r) be the mean velocity at RANS–LES interface known from the RANS solution. Then the velocity field u(r, t) imposed as the inflow boundary condition for LES at this interface is defined as follows: u(r, t) = U(r) + u (r, t),

(1)

where u (r, t) is the field of velocity fluctuations (“synthetic turbulence”). Similar to other methods of the same type (e.g., [4]), u (r, t) is defined so that the corresponding second moment tensor u i u j is equal to the Reynolds stress tensor R known from the RANS solution. This is reached by using Cholesky decomposition of the Reynolds stress tensor R = AT A. Then the synthetic velocity fluctuations in (1) can be defined via elements of the tensor A as u i (r, t) = ai j (r)v j (r, t), where vj (r, t) is the auxiliary field of the velocity fluctuations satisfying v j = 0 and vi v j = δi j . Thus the problem of definition of u (r, t) in (1) reduces to

definition of the v (r, t) field. In the present work, this field is prescribed in the form of superposition of weighted Fourier modes: N √ 6 n n n n nt n q σ cos k d · r + φ + s v (r, t) = 6 τ

(2)

n=1

Here: N is the number of modes, which is defined during the computations (see below); q n is the normalized amplitude of the n-th mode defined by the local energy spectrum; k n is the wave number of the n-th mode; dn is the random wave vector direction uniformly distributed over unit sphere; σ n is the unit vector normal to dn , and the angle defining its direction in the plane is a random number uniformly distributed in the interval [0, 2π ); φ n is the phase of the n-th mode, which is also a random number uniformly distributed in the interval [0, 2π ); s n is the non-dimensional frequency of the n-th mode with Gaussian distribution and the mean value and standard deviation equal to 2π ; τ is the global time-scale.

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Normalized amplitudes of the modes in (2) q n = E(k n )Δk n /

N

E(k n )Δk n ,

n=1

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qn = 1

(3)

n=1

are defined with the use of a modified von Karman spectrum (see Fig. 1): −17/6 E(k) = (k/ke )4 1 + 2.4 (k/ke )2 f η f cut

(4)

Here f cut and f η are empiric functions. The former provides damping of the spectrum in the vicinity of wave number corresponding to the Kolmogorov length-scale (it is designed based on the classic experiments of Comte-Bellot and Corsinn [2]) and the latter damps the spectrum for wave numbers larger than

the Nyquist one, kcut = 2π/lcut . The functions read as f η = exp −(12k/keta )2 and . / 1/4 (ν is f cut = exp − [4 max(k − 0.9kcut , 0)]3 /kcut ] , where kη = 2π/ ν 3 /ε the molecular viscosity, ε is the turbulence dissipation rate), and the wave length lcut

/ . is defined as lcut = 2 min max(h y , h z , 0.3h max ) + 0.1dw , h max with h y and h z being the local grid steps in the LES inflow section, h max = max(h x , h y , h z ), and dw is the distance to the wall. Finally, the wave number ke in (4) corresponding to the maximum of the spectrum E(k) is defined by wave length of the most energy-containing modes, le , of the synthetic velocity fluctuations or, in other words, by size of the most energycontaining eddies: ke = 2π/le . Note that a proper choice of le is of crucial importance for getting the velocity field rapidly evolving to the physically realistic one. In present work this length-scale is defined as follows: le = min (2dw , Cl lt )

(5)

where Cl = 3 is an empirical constant and lt is the length-scale of the turbulence 1/2 model used in RANS region (for instance, lt = kt / C μ ωt in the case if this is k − ω model). In the near-wall part of the flow (5) returns le equal to the doubled distance to the wall, whereas in the outer part of the boundary layer it reduces to the RANS length-sale. Examples of le (dw ) distributions in the canonic turbulent shear flows computed with the use of k − ω SST model [5] are presented in Fig. 2.

E(k)

E(k)~k–5/3 fcut fη ke

kcut

Fig. 1 Energy spectrum of the synthetic velocity fluctuation field

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Fig. 2 Distribution of the length-scales involved in (5) in a plane channel (a), ZPGBL (b), and plane free shear layer (c). 1 – le , 2 – lt , 3 – Cl lt , 4 – 2dw

A set of the wave numbers used in the turbulence generator (2) is common for the whole RANS–LES interface and forms geometric series k n = k min · (1 + α)n−1 , n = 1 ÷ N , α = 0.01 ÷ 0.05 (here k min = βkemin is the minimum wave number, β = 0.5, and kemin is the wave number corresponding to the maximum value of le : kemin = 2π/lemax , lemax = max {le (r)}). The value of N , i.e., the number of modes used in (2) is the maximum integer, for which k N satisfies the inequality k N ≤ kmax = 1.5 max {kcut (r)}. To finalize the formulation, we have to specify the time-scale τ in (2). It is defined via the quantity lemax and a macro-scale of the velocity in the interface section (e.g., the maximum or bulk velocity): τ = Cτ lemax /U , Cτ is the empiric constant. Note that exactly such global definition of the time-scale, coupled with the local scale of the energy-containing eddies le (5) results in forming of physically realistic (elongated in the streamwise direction) eddies in the inner part and nearly isotropic eddies in the outer part of the boundary layer.

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Fig. 3 Skin-friction distributions and mean velocity and Reynolds stresses profiles in the plane channel at Reτ = 400 predicted by hybrid algebraic WMLES model [6] with different inflow conditions (results with streamwise-periodic BCŠs are considered as a benchmark)

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3 Results The method outlined above has been applied to LES of three canonical shear flows: a channel flow at Reτ = 400, zero pressure gradient boundary layer (ZPGBL), and plane free shear layer. In all the cases the simulations were performed with the use of the algebraic hybrid WMLES model [6]. Results of the simulations shown in Figs. 3, 4 and 5 suggest that the inflow turbulent content created by the proposed method indeed ensures a rapid formation of realistic turbulent structures farther downstream: the length of relaxation from the inflow section to a mature LES solution turns out to be tangibly shorter than that with Synthetic Eddy Method (SEM) [4] currently considered as one of the best synthetic turbulence generators (see Fig. 3) and comparable to that with the recycling method [7] (see Fig. 4).

Cf 0.004 0.003

experiment SST RANS recycling present method

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The proposed method combined with the k −ω SST based Embedded IDDES [6] has also been applied to the flow past a wall mounted hump at Re = 936000 studied experimentally in [3]. Results of the simulation shown on Fig. 6 visibly demonstrate a significant improvement of the agreement with the experiment in the case of Embedded IDDES with the proposed synthetic velocity fluctuations at the RANS-IDDES interface compared to both RANS and IDDES in the whole domain. Acknowledgements This work was funded by the EU ATAAC (ACP8-GA-2009-233710) project, by Boeing Commercial Airplanes and, partially, by the Russian Basic Research Foundation (grant No. 09-08-00126).

References 1. Bechara, W., Bailly, C., Lafon, P., Candel, S.M.: Stochastic approach to noise modeling for free turbulent flows. AIAA J. 32, 455–463 (1994) 2. Comte-Bellot, G., Corrsin, S.: Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated, “isotropic” turbulence. J. Fluid Mech. 48, 273–337 (1971) 3. Greenblatt, D., Paschal, K., Yao, C.-S., Harris, J.: A separation control CFD validation test case part 2. Zero Efflux oscillatory blowing. 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, AIAA Paper 2005-0485 (2005) 4. Jarrin, N., Benhamadouche, S., Laurence, D., Prosser, R.: A synthetic-eddy-method for generating inflow conditions for large-eddy simulations. Int. J. Heat Fluid Flow. 27, 585–593 (2006) 5. Menter, F.R.: Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32, 1598–1605 (1994) 6. Shur, M., Spalart, P.R., Strelets, M., Travin, A.: A hybrid RANS-LES approach with delayedDES and wall-modelled LES capabilities. Int. J. Heat Fluid Flow. 29, 1638–1649 (2008) 7. Spalart, P.R., Strelets, M., Travin, A.: Direct numerical simulation of large-eddy-break-up devices in a boundary layer. Int. J. Heat Fluid Flow. 27, 902–910 (2006)

A Numerical Simulation of the Magnetically Driven Flows in a Square Container Using the Delayed Detached Eddy Simulation ˇ Vít Honzejk, and Kateˇrina Horáková Karel Frana,

Abstract The isothermal unsteady turbulent flow driven by a rotating magnetic field in a square container at Taylor number 1 × 106 was investigated numerically using in-house CFD code. The quasi isothermal electrically conducting fluid was exposed by the higher intensity of the magnetic induction, and consequently, the induced flow turned up to be turbulent at higher Reynolds or Taylor numbers. As a turbulent approach, the Delayed Detached Eddy Simulation model was successfully applied. The magnetic force calculation assumed the low frequency/low induction conditions. The effect of the rotating magnetic field gives rise to the time-independent magnetic body force, computed via the electrical potential equation and Ohm’s law and the time-dependent part that is neglected due to the low-interaction parameter. The characteristic results provided numerically were time-averaged velocity fields, turbulent quantities, vortex flow structures etc.

1 Introduction The use of the magnetic field for stirring of melts is a very attractive technique in metallurgy and single crystal growth processes introduced in e.g. [1]. The previous studies carried out in the last decays e.g. [4] demonstrated the possibility to control the heat and mass transfer applying the appropriate type of the magnetic field. As a consequence of this fundamental result is that the fluctuating natural convection evoked by the temperature differences can be suppressed partially or even completely using the effect of the magnetic field. Whereas the influence of the magnetic force on the electrically conducting fluid in the cylindrical axisymetric container was studied extensively in the last two decades, the less attention has been dedicated to the same physical problem in the non-axisymetric (cubic) containers. Furthermore, the higher complexity of this flow problem emerges if the turbulent flow regime K. Fraˇna (B) Department of Power Engineering Equipment, Technical University of Liberec, 461 17 Liberec, Czech Republic e-mail: [email protected]


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at higher Reynolds numbers are considered. Taking into account a turbulent flow regime, the turbulence has an effect on the flow behavior mostly due to stress tensors or viscosity term in the Navier–Stokes equations. Besides, the structure and intensity of turbulence can be strongly affected by the applied magnetic field as well. The interaction parameter N = σB02l/ρu describes the ratio of the eddy turnover time and the Joule time. Depending on this parameter value, several scenarios are possible. In our particular case, the interaction parameter N is in order of 10−4 so that N 0, such that the spectral content on the computational grid is limited to the range [0, κc ]. Then, an optimal value for ck,n may be found, for which a norm of the magnitude of (5) is minimal in the spatial domain. This is equivalent of minimizing the dispersion error of the resulting central finite difference scheme, weighted with the energy spectrum of u (x), over the interval [0, κc ]. Such a value of ck,n , generally ∗ , for results in a finite difference approximation of order O Δk , unless ck,n = ck,n k+2 which it yields O Δ . This approach, which is equivalent to the DRP approach, is more advantageous than increasing the order of accuracy a priori [4]. It was shown in [2] that the optimal coefficient ck,n can be determined by minimizing the difference between the truncated equations (3) and (4). Subtracting (3) and (4) gives

E = L + ck,n M = O (αΔ)k − O Δk .

(6)

The terms L and M, which are found by identification in the former subtraction, can be further simplified [2], yielding

On the Performance of Optimized Finite Difference Schemes in Large-Eddy Simulation

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Δ

δ k+n u ∗ L = ck,n (7) α k − 1 Δk δx k+n K

k+n u Δ k+n+2 u Δ k k δ k k ∗ 2 2 δ M = 1−α Δ , (8) − α Δ f c2,k+n 1 − α Δ δx k+n δx k+n+2 ∗ is a constant coefficient known from the Taylor series expansion. The where c2,k+n from (6) by minimizing E using a optimal coefficient ck,n can then be determined least squares approach, i.e. ∂c∂k,n E 2 = 0, where · denotes an averaging operadyn LM tor, resulting in the dynamic coefficient ck,n = − MM . Here, only global spatial dyn averaging is considered for the finite difference schemes. Once ck,n is calculated, its value can be used in the finite difference approximation (3). However, since an explicit finite difference approximation for the (k + n) th derivative is used, the stencil of (3) is then strictly required for this implicit scheme. This is remedied by substituting the explicit (k + n)th derivative by an implicit formulation, which is equivalent of writing (3) immediately in its most compact formulation. We finally obtain the compact implicit discretization q l=−q

4 3 dyn dyn 4 r

nu ck,n c β ∂ j k,n u x ≈ αl − ∗ αl − αl 1 − (x ) i+l i+ j ∗ ck,n ∂xn Δn ck,n

3

j=−r

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∗ ck,n

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where αl and β j denote the weighting coefficients of the k + 2nd-order implicit finite difference approximation. The resulting dynamic scheme (9) has a formal dyn ∗ , which then leads to order k + 2. order of accuracy k unless ck,n = ck,n dyn ∗ If f = 0, ck,n = ck,n . Expression (9) is then O Δk+2 . If u (x) is smooth, i.e. π κc /κmax → 0, κmax = Δ , then E ≈ 0. Applying a small perturbation analysis on this expression [2], showed that for κc /κmax → 0, f approaches an asymptotic value f ∗ , which can be determined analytically. If κc /κmax , 0, an optimal value of f is determined by calibrating the modified wavenumber κn n (κ) of (9), for a turbulent spectrum at Re → ∞ with fixed filter-to-grid cutoff ratio κc /κmax , such that the dispersion errors are minimal in the range 0, κc /κmax . The obtained value of f guarantees that the dynamic scheme reaches maximum performance for a Large-Eddy Simulation at high Reynolds numbers with a maximum filter-to-grid cutoff ratio κc /κmax . The optimal value for the blending factor f is calculated by solving ∂ ∂f

0

π Δ

2 n n κ − κn (κ, f ) E u (κ) dκ = 0,

(10)

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Table 1 Numerically obtained optimal blending factors f and corresponding values of ck,n for the inertial range model spectrum at κc = 23 κmax Slope

Scheme

n=1 f

ck,1

n=2 f

ck,2

β = − 53

Explicit k = 2 Explicit k = 4 Implicit k = 4

0.2555 0.2298 0.1241

−0.3088 0.0740 0.0121

0.2339 0.2241 0.1363

−0.1310 0.0203 0.0069

β = − 73


0.2654 0.2329 0.1261

−0.2966 0.0724 0.0120

0.2353 0.2248 0.1370

−0.1293 0.0201 0.0069

β=0


dyn

−0.3344 0.0775 0.0119

dyn

−0.1345 0.0206 0.0069

where E u (κ) represents the energy spectrum of the flow field u (x). An idealized inertial range scaling is assumed, i.e. E u (κ) = [1 − H (κ − κc )] κ −β =

κc

(11)

where β determines the slope of the inertial range and the cutoff wavenumber κc indicates the highest appearing wavenumber in the (resolved) field u (x). We take β = −5/3 for the turbulent velocity u (x), and β = −7/3 for the turbulent pressure field p (x). Table 1 gives on overview of the blending factors and the corresponding values of the dynamic coefficient for κc = 23 κmax , for the 2nd- and 4th-order explicit and the 4th -order implicit DFD schemes. For β = 0, DRP schemes are obtained.

2 Taylor–Green Vortex The Reynolds number of the TG-vortex is Re = 1,500, which corresponds to a transversal Taylor micro-scale Reynolds number Reλ ≈ 55 [1]. The DNS-solution serves as a reference solution against which the various LES-solutions are compared. The system of equations is directly solved on a uniform computational grid with 2563 nodes (1283 Fourier modes), which is sufficient for this Reynolds number [1]. The Large-Eddy Simulations are performed on a uniform computational grid with 643 nodes and with grid cutoff wavenumber κmax = π/Δ = 32. The LES equations are solved in the double-decomposition framework. This implies that the nonlinear term and the residual-stress tensor τ i j = u i u j − u i u j are filtered explicitly with a sharp cutoff filter. Two residual stress models for τ i j are considered. The dynamic Smagorinsky model τ i j = −2νe S i j = −2Cs2 Δ2c S S i j and Cs2 is determined by the dynamic Germano procedure. The small-small mul

2 Δ2 S S tiscale Smagorinsky model τ i j = −2 νe Sij = −2C s,m in which c ij


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−3/4 (.) , denotes the band-pass filter [λc , κc ] and C s,m = Cs γ γ 4/3 − 1 , with γ = κc /λc and C s ≈ 0.17. The cutoff wavenumber λc that determines the secondary sharp cutoff filter in the dynamic procedure or the sharp cutoff scale-separation filter in the multiscale model is determined as λc = κc /2 = κmax /3. For the DNS and the spectral LES, a pseudo-spectral method is used. The skewsymmetric formulation is adopted for the discretization of the nonlinear term such that it conserves the kinetic energy. Only the residual stress and the molecular viscosity are dissipative. 4-stage Runge–Kutta method is used with A low-storage 1 1 1 standard coefficients 4 , 3 , 2 , 1 . A sufficiently small time step Δt = 0.005 was chosen for both the DNS and the LES, such that the dispersion and dissipation errors related to the time-stepping remain sufficiently low. The solution is advanced in time up to t = 14.25, where the maximum dissipation occurs around t = 9. For the Large-Eddy Simulations with the DFD schemes, each partial derivative in the Navier–Stokes equations or the Poisson equation is discretized by the appropriate dynamic finite difference approximation. The implementation involves the calculation of 36 dynamic coefficients which can be evaluated at each Runge–Kutta step, that is 4 times per time step. To reduce computational overhead, the dynamic coefficients are evaluated every 10th time step, which is expected to be sufficient. The computational overhead is only 1.7% in comparison with DRP schemes of comparable accuracy.

3 Numerical Errors and Modeling Errors The error decomposition method of Vreman et al. [5] and Meyers et al. [3] is adopted. For a parameter φ, the total error is decomposed into a modeling error contribution, defined as the difference between φ in the DNS and in the spectral LES, and a numerical error, defined as the difference between φ in the spectral LES and the finite difference LES. The global error norms of three quantities are analyzed: the longitudinal integral length scale L 11 , the kinetic energy k, and the dissipation rate ε. In order to obtain a single-value error norm instead of an error that varies in time, we integrate the previously defined error norms in time. Figure 1 shows the time-integrated numerical and total error norms. The numerical error-norms decrease with increasing order of accuracy, for all scheme-types, i.e. standard, DRP and DFD. A priori optimization of finite difference stencils in the DRP schemes, leads to a modest improvement of the numerical accuracy, in comparison with standard FD methods with the same stencil width. Real-time optimization of the stencils, such as in the DFD schemes, leads to a larger reduction of the numerical error. Indeed, the DFD schemes are able to adjust their stencil coefficients, by minimizing the numerical error at each instant of the simulation, taking the instantaneous shape of the energy spectrum into account. Hence, the scheme has different spectral characteristics in the laminar and the turbulent flow regimes. However, a better numerical accuracy does not necessarily lead to a smaller total

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error, as demonstrated in Fig. 1. For instance, increasing the order of accuracy of the standard schemes tends to result in an increased total error. Moreover, the total errors of the DRP and DFD schemes, are even larger, despite their good numerical performance. It was found that both eddy-viscosity models are too dissipative, resulting in a positive sign of the modeling error. Note that the modeling error was found to be dominant in the simulation. The numerical errors of the standard asymptotic finite difference schemes have a negative sign, which indicates a significant reduction in the dissipation due to numerics, whereas, the DRP and DFD schemes lead to a positive sign of the error, indicating an increased dissipation due to the numerics. As a consequence, cancellation of numerical errors and modeling errors are witnessed for the standard finite difference schemes, due to their opposite signs. This confirms the results in [3].

References 1. Brachet, M.E., Meiron, D.I., Orszag, S.A., Nickel, B.G., Morf, R.H., Frisch, U.: Small-scale structure of the Taylor–Green Vortex. J. Fluid. Mech. 130, 411–452 (1983) 2. Fauconnier, D., De Langhe, C., Dick, E.: Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor–Green vortex. J. Comput. Phys. 228(21), 8053–8084 (2009)


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3. Meyers, J., Geurts, B., Baelmans, M.: Database analysis of errors in large-eddy simulation. Phys. Fluids 15(9), 2740–2755 (2003) 4. Tam, C.K.W., Webb, J.C.: Dispersion-relation preserving finite difference schemes for computational acoustics. J. Comp. Phys. 107, 262–281 (1993) 5. Vreman, B., Geurts, B., Kuerten, H.: Comparison of numerical schemes in large-eddy simulations of the temporal mixing layer. Int. J. Num. Methods Fluids 22, 297–311 (1996)

Part XXVII

Combustion/Supersonic Flow

Wide-Range Single Engine Operated from Subsonic to Hypersonic Conditions: Designed by Computational Fluid Dynamics Ken Naitoh, Kazushi Nakamura, Takehiro Emoto, and Takafumi Shimada

Abstract A new type of single engine capable of operating over a wide range of Mach numbers from subsonic to hypersonic regimes is proposed for airplanes. Traditional piston engines, turbojet engines, and scram engines work only under a narrower range of operating conditions. The new engine has no compressors or turbines such as those used in conventional turbojet engines. A notable feature is its system of super multijets that collide to compress gas for the transonic regime. A numerical model simulating compressible turbulence with chemical reactions based on the CIP and BI-SCALES methods is employed to design the engine. The maximum power of this engine will be sufficient for actual use. For the higher Mach numbers in supersonic and hypersonic conditions, this engine can take the mode of a ram or scramjet engine.

1 Introduction Traditional engines such as piston engines, turbojet engines, ram engines, and scram jet engines operate only in a narrow range of Mach numbers. Piston engines work well in the subsonic regime and turbojet engines are also suitable around the transonic regime, whereas ram and scram jet engines can be employed only for supersonic or hypersonic conditions. Although combinations of traditional engines can theoretically propel aircraft beyond the Earth’s atmosphere, such aircraft would be too heavy and complex to use. Rocket systems, on the other hand, require too much fuel and have safety issues. The key question is whether or not the compressors and turbines of turbojets systems can be eliminated [1].

K. Naitoh (B) Faculty of Science and Engineering, Waseda University, Shinjuku, Tokyo 169-8555, Japan e-mail: [email protected]


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2 Outline of the New Single Engine For low Mach number conditions, a new pulsed compression concept is achieved by jetflows that collide at a single point after the airflow enters from super multiple side passages I2 and nozzles for super multijets C2. (See the combustion area in Fig. 1.) After combustion occurs around the collision point of the jetflows, the side passages generating these super multijet flows are alternately closed by means of a rotating flat plate C4. The outflow from each side passage pulsates because small holes provided in the rotating flat plate alternately direct flow to the side passages. Because the rotating plate is flat, less energy is needed for rotating it. The super multiple side passages are completely closed for higher Mach numbers. Under the condition, the main intake passage I1 located in front of the super multijet nozzles, takes in air more. That results in a ram or scramjet engine for supersonic and hypersonic conditions. (See the combustion area in Fig. 1.) As the jet nozzles are covered by the smooth shape of the wings I3, the area of the main intake passage I1 gradually changes along the engine axis, which prevents separation flows (Fig. 2). (Accordingly, the main intake passage can also have the effect of Laval nozzle.) We examined the potential of this engine by using numerical simulations. Threedimensional compressible Navier–Stokes equations were solved with a simplified two-step chemical reaction model. The cubic interpolated pseudo-particle (CIP) method [6] applied for the governing equation of the multi-level formulation [2–5] was employed, which is suitable for calculating both compressible and incompressible problems. The peak pressure at the combustion center was found to be over 2.5 MPa, while that just before ignition was over 1.0 MPa (Fig. 3). The maximum power of this engine will be sufficient for actual use (Table 1).

Fig. 1 Wide-range single engine operating from subsonic to hypersonic regimes [1]

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Fig. 2 Super multijet nozzles generating compression and the wings covering the jets

Fig. 3 Time history of pressure at the center of combustion area Table 1 Power performance of the new engine

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Fig. 4 Time evolution of the temperature distribution during one cycle of detonation

Figure 4 shows the computed temperature distribution in the combustion chamber surrounded by seventeen nozzle jets during one cycle in a transonic regime. Figure 5 presents the instantaneous stream lines of the exhausted flow.

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Fig. 5 Instantaneous streamlines in the combustion area

3 Advantages of This Engine (a) There are no mechanical components around the center axis of the engine, because of the combustion point produced by the super multiple jets coming from the side passages. Then, the axisymmetric engine design means the higher temperature region of burned gas is far from the solid walls of the engine. This geometry markedly reduces the heat loss at the walls, to substantially improve efficiency and power. (b) In turbojet engines, the burned gas temperature is limited by reliability of the turbine blades. However, the proposed engine permits higher temperatures, i.e., higher compression ratios before combustion. This also leads to better efficiency and higher power.

4 Engine Start The engine system shown in Fig. 1 cannot work at very low Mach numbers less than 0.1 or under starting conditions. To overcome this issue, we propose an ultimate engine system that can also be used at engine start (Fig. 6). This ultimate system is the extended version fitted with a special piston system having an exhaust duct and a rotating plate for closing the duct (twister system). The engine system is practicable, because both the engine with super multijets and the piston engine are of the pulsating type, which should allow a smooth transition between the two. A combination of turbojet and piston engines would be difficult to achieve.

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Fig. 6 Wide-range single engine operating from startup to hypersonic conditions [1]

5 Noise Level The vapor fuel distribution inside the combustion chamber can be changed by varying the fuel injectors and their injection timings. Figure 7 shows the influence of fuel distribution patterns on pressure time histories. Optimization of the fuel distributions will lower the peak pressure after combustion, thereby reducing the noise level, while the high power is kept.

Fig. 7 Pressure time histories for different vapor fuel distributions. (A higher pressure peak for a center charge and two lower ones for a surrounding charge of fuels)

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6 Conclusion The proposed engine capable of operating from startup to hypersonic conditions promises to bring a new transport mode. The automobile mode with the piston engine can be used from startup to takeoff from highway, while the pulsating mode of colliding super multijets works around the transonic regime and the steady flow mode of ram and scram jets can be employed for hypersonic conditions. First, we want to develop a small passenger vehicle system for ground and air, at a price of equal to that of luxury automobiles with a 5-liter engine. The present research on the new engine may represent the first quantum leap in engineering fields achieved by using only computational fluid dynamics. We will shortly enter an age of “computer-aided prototyping (CAP)” for achieving several quantum leaps.

References 1. Naitoh, K.: A new cascade-less engine operated from subsonic to hypersonic conditions. J. of Thermal Sci. 19(6), 481 (2010) 2. Naitoh, K., Kuwahara, K.: Large eddy simulation and direct simulation of compressible turbulence and combusting flows in engines based on the BI-SCALES method. Fluid Dyn. Res. 10, 299–325 (1992) 3. Naitoh, K., Nakagawa, Y., Shimiya, H.: Stochastic determinism approach for simulating the transition points in internal flows with various inlet disturbances. Proceedings of the ICCFD5 on Computational Fluid Dynamics 2008, Springer, Heidelberg (2008) 4. Naitoh, K., Shimiya, H.: Stochastic determinism capturing the transition point from laminar flow to turbulence. Japan Journal of Industrial and Applied Mathematics [Also in Proceedings of 6th International Symposium on Turbulence and Shear Flow Phenomena (TSFP6), 2009] 28(1), 3–14 (2011) 5. Shimada, T., Naitoh, K., Nakamura, K., Emoto, T.: Computation of aero-craft engine based on supermulti jet. Proceedings of the 23rd conference on CFD, Sendai (2009) 6. Takewaki, H., Nishiguchi, A., Yabe, T.: The cubic-interpolated pseudo-particle (CIP) method for solving hyperbolic-type equations. J. Comput. Phys. 61, 261 (1985)

Numerical Simulations of the Performance of Scramjet Engine Model with Pylon Set Located in the Inlet I.M. Blankson, A.L. Gonor, and V.A. Khaikine

Abstract This work is devoted to numerical simulations of the flow in the scramjet engine model with hydrogen/air mixture combustion. The sub-scale model consists of the inlet with pylons, fuel injectors located on pylons, mixing region, combustion chamber, and nozzle. The goal of this work is to obtain efficient and stable combustion in the combustion chamber and thrust using this design, and to compare it with other existing scramjet engine models. A pylon set is proposed for installation in the rectangular inlet to decrease drag of the inlet and to create several air/fuel mixing layers to increase mixing efficiency. A movable cylindrical rod was placed in the combustor to initiate shock-induced combustion. Numerical simulations were conducted using the NASA CFD code VULCAN and performance of this scramjet engine model was computed. These simulations showed that efficient combustion and resulting thrust addition in the case of this design significantly exceeds the drag introduced by the obstacle in the combustor. As a result, in the relatively short combustor, we obtained the combustion efficiency, ηc = 0.85, and a relatively high value of the engine model specific impulse, Isp = 1,275 s.

1 Geometry of the Proposed Scramjet Engine Model Based on the results on the He/Air mixing efficiency in the inlet with pylons [1], we have investigated flow and combustion of the air/fuel mixture in the model scramjet engine. Such engine is built of a relatively short inlet with pylons, combustor, and nozzle (Fig. 1). The inlet and combustion chamber are two-dimensional, while the nozzle has three-dimensional configuration with flat surfaces and rectangular cross section. The sizes of the scramjet engine were chosen so that the experimental testing of this model in hypersonic wind tunnels were possible. It is to be noted, that this scramjet model is significantly smaller than the CIAM/NASA scramjet model [8], well-known X-43 model [3, 4, 9], and the Japan scramjet model [6, 7, 10]. In the I.M. Blankson (B) NASA Glenn Research Center, Cleveland, OH 44135, USA e-mail: [email protected]


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2 CFD Analysis of the Scramjet Model Performance Numerical simulations of the flow with combustion in the channel and engine nozzle were conducted using the NASA CFD code VULCAN. The analysis of the temperature distribution showed that the temperature of the air/fuel mixture in the channel is not high enough for the intensive and stable combustion to occur. Due to this, it was suggested to place a cylindrical obstacle (cylindrical rod) at the entrance of the combustion chamber. This cylindrical rod will favor the increase in the temperature in the area of combustion triggering (Fig. 1). The Mach number distribution near the obstacle and in the combustion chamber is shown in Fig. 2. In the immediate region after the obstacle, in its trace, according to Fig. 2 and according to the Mach number distribution in a cross section (Fig. 3), there exists a subsonic region, which allows the possibility for the subsonic reactive flow. The configuration of the head shock wave in front of the cylindrical rod proves that, first, the flow field in the channel is non-uniform, and, second, a shock-induced combustion occurs near the head shock wave, which (under the certain conditions) can transform into detonation combustion. The latter also agrees with the temperature field distribution shown in Fig. 4 where the bow shock wave is accompanied by the narrow zone of high temperature and with the high values of the mass fraction H2 O shown in Fig. 5. The temperature peak, according to Fig. 6, occurs in the obstacle trace, and the stream in the channel intensively combusts.

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to die out, shows that there exists a small excess of hydrogen (approximately 15%) which remains unburned. Obviously, this is due to the non-uniform distribution of the flow parameters in channel cross sections. However, in general, the combustion in the channel is stable and is accompanied by the significant heat generation. The combustion efficiency, ηc , was defined [6] as the ratio between the H2 consumption rate and the H2 supply rate. It can be presented as: H

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nozzle entrance. It turned out that using this assumption, combustor length equals only 6.8% of the total engine length. Such short combustor became possible due to the intensive combustion that takes place in its channel. It is to be noted, that these results on the intensification of combustion became possible only because of the movable cylindrical obstacle which was introduced and added into the engine design. The optimum location of this cylindrical rod was determined using numerical simulations results. These numerical simulations showed that the thrust addition in this case significantly exceeds the drag introduced by the obstacle. The calculation of the force characteristics of different engine components and of the total engine design gives the following results: Drag: Pylons (3) – 0.81 kg; Forebody and channel with combustor (including side walls) – 8.78 kg; Cylindrical obstacle – 4.3 kg; Thrust: Nozzle – 34.91 kg; Net engine model thrust – 21.02 kg. One should note that previous numerical simulations and experimental tests at Mach number 4 [5] showed that pylons, located at the inlet entrance (without fuel injection), produced additional thrust. However, in the current design with fuel injectors located on pylons, a small additional drag occurs, which one can relate to the fact that additional shock waves are produced by the fuel jets. Nevertheless, there is a possibility to improve pylons thrust by replacing existing inlet forebody with a two-stage wedge or with a wedge that transforms partial into a curvilinear surface. Such shape leads to isentropic compression and turns the flow by a bigger angle compared to the current shape with a single wedge. As a result the drag and total pressure losses, caused by the main shock wave, will decrease, and pylons are placed at a bigger angle to the flow, which increase their thrust. Now let us estimate the efficiency of the engine model using the usual criterion -ratio of the total engine thrust to fuel flow weight per unit time. This value may be represented as:

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by using (1) for engines mentioned above [6, 7, 10], equals Isp = 850 s, for the equal values of equivalent ratio, = 1, corresponding to the maximum thrust. The relatively high value of Isp obtained for the scramjet model with movable rod is rather unexpected since the cylindrical rod induces additional shock and losses of total pressure. To clarify this situation we used 1-D approach presented in [2]. The difference of total pressure in different points of 1-D flow containing shocks, heat inflow, q, through the outer boundary or from chemical reactions, and uncompensated heat inflow, q , caused by the irreversible (viscous) processes may be represented as follows, x1 p0 f − p 0 = x0

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where subscript 0 relates to stagnation values of the parameters, qx = dq/dx and qx = dq /dx. The heat inflows q and q are given per unit mass and unit length, respectively, δp0sh is the difference between total pressure values in front of the shock and behind it. According to Eq. (3), movable rod placed in the combustor increases δp0sh and losses of the total pressure in the flow. On the other hand, the first item on the right hand side of (3) may be in this case smaller, compared with the one for combustor with diffusion combustion. The fact is that intensive detonation type combustion occurs at short distances, Δx, from shock, that is q x = 0 only at x ∈ (x 0 , x0 + Δx) with T ∼ T0 . In the case of slow diffusion combustion the values of qx = 0 occur within long interval along the flow whereas combustion transpires at T < T0 . As a result, the overall losses of the total pressure in the scramjet design with rod can turn out to be smaller than in the case of the scramjet based on the diffusion combustion, since the reduction of first and second items in (3) may offset the increase of the third item.

3 Conclusions New model of scramjet engine with pylons located in the inlet and movable rod located in the combustor was designed. Numerical simulations of this scramjet engine model were conducted and it was shown that in spite of the non-uniform flow field and non-uniform air/fuel mixture, suggested design allows to obtain intensive combustion which takes place in the whole volume of the combustor channel, and to generate thrust. Comparison of this model with other scramjet engine models available in literature showed that even with no optimization of major parameters of the given engine model, the performance of this design according to conducted numerical simulations is promising. The specific impulse of the proposed scramjet engine model, Isp = 1, 275 s.

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References 1. Akyurtlu, A., Akyurtlu, J., Gonor, A.L., Khaikine, V., Cutler, A.D., Blankson, I.M.: Numerical and experimental tests of a supersonic inlet with Pylon Set and Fuel Injection through Pylons. AIAA 44th Aerospace Sciences Meeting, Reno, Nevada, USA. AIAA Paper 2006-1032 (2006) 2. Chernyi, G.G.: Gas Dynamics. CRC Press, New York, NY (1994) 3. Ferlemann, P.G.: Comparison of Hyper-X Mach 10 scramjet preflight prediction and flight data. AIAA/CIRA 13th International Space Planes and Hypersonic Systems and Technologies Conference, Capua, Italy. AIAA 2005-3352 (2005) 4. Ferlemann, S.M., McClinton, C.R., et al.: Hyper – X Mach 7 Scramjet design, ground test and fight results. AIAA/CIRA 13th International Space Planes and Hypersonic Systems and Technologies Conference, Capua, Italy. AIAA 2005-3322 (2005) 5. Gilinsky, M.M., Khaikine, V., Akyurtlu, A., Akyurtlu, J., Blankson, I.M., et al.: Numerical and experimental tests of a supersonic inlet utilizing a pylon set for mixing, combustion and thrust enhancement. AIAA/CIRA 13th International Space Planes and Hypersonic Systems and Technologies Conference, Capua, Italy. AIAA 2005-3290 (2005) 6. Kouchi, T., Mitani, T., Masuya, G.: Numerical simulation in scramjet combustion with Boundary layer bleeding. AIAA J. Propulsion and Power 21, 4 (2005) 7. Mitani, T., et al.: Boundary-layer control in Mach 4 and Mach 6 scramjet engines. AIAA J. Propulsion and Power 21, 4 (2005) 8. Rodriguez, C.G.: Computational fluid dynamics analysis of the Central Institute of Aviation Motors/NASA Scramjet. AIAA J. Propulsion and Power 19, 4 (2003) 9. Rogers, R.S.: Scramjet development tests supporting the Mach 10 flight of the X-43. AIAA/CIRA 13th International Space Planes and Hypersonic Systems and Technologies Conference, Capua, Italy. AIAA 2005-3351 (2005) 10. Tomioka, S., et al.: Distributed fuel injection for performance improvement of staged supersonic combustor. AIAA J. Propulsion and Power 21, 4 (2005)

Scale Adaptive Simulations over a Supersonic Car Guillermo Araya, Ben Evans, Oubay Hassan, and Kenneth Morgan

Abstract In this study, the unsteady Reynolds-averaged Navier-Stokes equations are employed together with the Menter SST-SAS turbulence model in compressible flows. Numerical simulations over a supersonic car, the BLOODHOUND SSC (http://www.bloodhoundssc.swan.ac.uk/), are shown and discussed with Mach numbers up to 1.3.

1 Introduction Computational fluid dynamics (CFD) has experienced a notable growth in the last few decades. Nowadays, most engineering designs and technical projects rely on computational predictions before making critical design decisions; additionally, CFD may also be employed to gain important insight into the flow physics before performing an expensive experiment. Recently, significant attention has been paid to relatively low-cost (compared to large eddy simulations, LES) time-dependent computations of complex flows for industrial applications, e.g. geometries with moving parts, wing flutter, noise prediction, etc. Particularly, the unsteady Reynoldsaveraged Navier–Stokes (URANS) methodology has became quite popular, due to its successes in predicting the most energetic modes or coherent structures. However, URANS has frequently been accused of inaccurately representing the correct spectrum of turbulent scales, even if the numerical grid and the time step would be of sufficient resolution. In this study, the URANS equations for compressible flow are solved over a supersonic car, the BLOODHOUND SSC [1]. The Menter SST turbulence model is employed in conjunction with a recently developed SAS model (scale adaptive simulations) by Menter et al. [2]. This allows capturing more details of the flow or the “small turbulent scales”. The FLITE flow solver [3] developed at Swansea University is applied, based on a finite volume approach with stabilization and discontinuity capturing. Modelling the aerodynamics of a supersonic car is very challenging. This is due to the presence of highly separated flow regions G. Araya (B) Civil & Computational Engineering Centre, Swansea University, Swansea SA2 8PP, UK e-mail: [email protected]


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and shock waves; not to mention the numerical modelling of the rotating wheels and accounting for the car–ground interaction. The aerodynamic performance of the BLOODHOUND SSC, as well as the most relevant aspects of the flow physics will be analyzed by using the Menter SST-SAS model in this study.

2 Numerical Results The Swansea FLITE3D flow solver [4], which employs unstructured meshes and a finite – volume approach, is used in the assessment of the Menter SST-SAS turbulence model. More details about the numerical code can be found in [6]. The results of classical aerodynamic test cases are presented in this section, together with a discussion concerning the application to the BLOODHOUND supersonic car.

2.1 Acoustic Cavity Numerical simulation of flow over a 3-D rectangular cavity are presented. The cavity configuration is selected as the M219 experimental test case of Henshaw [5]. The geometry dimensions of the M219 cavity are L × W × D = 5 × 1 × 1 (length, width, and depth), with a depth D of 4 inches. The freestream Mach number, M∞ , is 0.85 and the Reynolds number is 19.6×106 based on the cavity depth. The hybrid mesh consisted of around 3.37 million tetrahedral elements, 4,472 prisms and 459 pyramids. The mesh has 20 viscous layers for boundary layer capturing and the first off wall point is located at y/D = 2.5 × 10−6 . The selected normalized time step is Δt ∗ = Δt/(D/U∞ ) = 0.05. The time variation of the total drag over the cavity, normalized by the reference surface and freestream dynamic pressure, can be observed in Fig. 1. Additionally, Fig. 2 shows iso-surfaces of Ω 2 − S 2 , where Ω is the vorticity and S 2 is the scalar invariant of the strain rate tensor. This parameter represents the large scale turbulence structures of the flow and qualitatively very similar structures were obtained by [3] in the cavity.

2.2 ONERA M6 Wing In simulation of flow over the ONERA M6 wing [6], the freestream Mach number M∞ = 0.84 and the total Reynolds number is 12×106 , based on the mean geometric chord. This case is tested at an angle of attack α = 6.0o . This case at the maximum angle of attack is very challenging for the turbulence models, due to the presence of highly – separated flow. A hybrid mesh is employed, consisting of around 6.3 million tetrahedral elements, 0.25 million prisms and 11.6 thousand pyramids. The wing surface is represented using around 1.1 million triangles. The mesh has 35 viscous layers for boundary layer capturing and the first off wall point is located at y + ≈ 0.4 in wall units, for points located in the vicinity of the leading edge, where

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2.3 F15 Fighter The third example consists on the generic F15 fighter configuration. The freestream Mach number is M∞ = 0.9, the total Reynolds number is 358×106 and the angle of attack is five degrees. The hybrid mesh is composed by around 2.2 million nodes and 10.2 million elements. In Table 1, the corresponding values of the pressure coefficients, C p , are exhibited for the upper and lower surfaces at three different points in the upper and lower surfaces (labeled as a,b and c in Fig. 4). The computed C p are compared with experimental data collected in wind-tunnel and in flight [7]. In general, the comparison of present results with experimental data is fair. The most significant discrepancies were found in point b of the lower surface, which may indicate a poor resolution of the mesh in this zone or an insufficient sample for statistics computation; and, further investigation must be done. Figure 5 shows isosurfaces of the instantaneous vorticity. Clearly, the appearance of large turbulence scales can be appreciated due to the SAS model.

2.4 Supersonic Car Preliminary simulations have been undertaken of unsteady flows over the supersonic car, BLOODHOUND SSC, at a free stream Mach number of M∞ = 1.3. The total Reynolds number is approximately 390×106 based on the stream wise length (L ∼ 13 m) of the car. One of the latest configurations (10b) has been considered, with a hybrid mesh of approximately 14 million elements and 3.12 million nodes. In Fig. 6, the time variation of the total drag over the car up to a non-dimensional time, Table 1 Pressure coefficients C p in F15 at α = 5o Upper surface

Lower surface

Point

Present C p

Flight data

Wind-tunnel data

Present C p

Flight data

Wind-tunnel data

a b c

−0.33 −0.15 −0.11

−0.24 −0.35 −0.1

−0.35 −0.2 −0.08

−0.36 −0.24 −0.023

−0.62 −0.01 −0.015

−0.42 −0.083 0.0034

(a)

(b)

Fig. 4 Iso-contours of pressure coefficient in F15 fighter. (a) Upper surface, (b) lower surface

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VORT: 0

0.003

0.007

0.03

0.07 0.1 0.3 0.5 0.7 1

5

50 100 500 1000 2000

Fig. 5 Iso-surfaces of instantaneous vorticity in the F15 fighter 6

Total Drag / (q x S)

5 4 3 2 1 0 –1

0

10

20

30

40

50

t*= t / (D / U∞) Fig. 6 Time variation of the total drag in the supersonic car

t ∗ , of 28 can be observed. Despite the relatively short calculated physical time, some important conclusions can be drawn from numerical results so far. Figure 7 depicts the iso-contours of the streamwise velocity at t ∗ = 28 at the half cross-sectional plane of the supersonic car. The vertical iso-lines at the entrance of the intake duct


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Fig. 7 Iso-contour of instantaneous streamwise velocity at the half cross-sectional plane

suggests the presence of a normal shock. Inside the duct, the flow experiences a slight deceleration from approximately a normalized velocity of 0.8–0.6 at the inlet turbofan. Two zones of highly accelerated flow and high lift are observed over the car: at the intake duct and over the horizontal tail. In addition, both engine outlets (turbofan and hybrid rocket) represent important sources of vorticity, as seen in Fig. 8. Finally, in Fig. 9 iso-surfaces of the parameter Ω 2 − S 2 are depicted. It can be observed the large turbulence structures flowing out downstream from the wheels. Although the selected wheel configuration might increase significantly the shape drag, this design allows the formation of oblique shocks well far from the wheel-ground interface.

Fig. 8 Iso-contour of instantaneous vorticity at the half cross-sectional plane in the car rear

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Fig. 9 Iso-surfaces of Ω 2 − S 2 in BLOODHOUND SSC

3 Conclusions An evaluation of the Menter SST-SAS turbulence model in URANS of turbulent compressible flows is performed for a number of 3D aerodynamic test cases (acoustic cavity, ONERA M6 wing and F15 fighter) by means of the Swansea FLITE3D flow solver. Furthermore, preliminary numerical predictions of the flow over the BLOODHOUND supersonic car [1] have also been carried out and important insights were acquired on the most significant aspects of flow phenomena. Acknowledgements The authors acknowledge the provision of Hector supercomputer resources by EPSRC under project numbers e147 and e160.

References 1. http://www.bloodhoundssc.swan.ac.uk/ 2. Henshaw, M.J.: M219 cavity case. In: Verification and Validation Data for Computational Unsteady Aerodynamics, pp. 453–472. Technical Report RTO-TR-26, AC/323/(AVT) TP/19, East Riding of Yorkshire, UK (2000) √ 3. Menter, F.R., Egorov, Y., Rusch, D.: Steady and unsteady flow modelling using the k − kL model. In: Turbulence, Heat and Mass Transfer 5, Proceedings of the International Symposium on Turbulence, Heat and Mass Transfer Dubrovnik, Croatia, 25–29 September 2006


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4. Morgan, K., Peraire, J., Peiro, J., Hassan, O.: The computation of 3-dimensional flows using unstructured grids. Comput. Methods Appl. Mech. Eng. 87, 335–352 (1991) 5. Schmitt, V., Charpin, F.: Pressure Distributions of the ONERA M6 Wing at Transonic Mach Numbers. Report AR–138, AGARD, Paris (1979) 6. Sørensen, K.A.: A Multigrid Accelerated Procedure for the Solution of Compressible Fluid Flows on Unstructured Hybrid Meshes. PhD thesis, University of Wales, Swansea (2002) 7. Webb, L., Varda, D., Whitmore, S.: Flight and Wind-tunnel Comparisons of the Inlet/Airframe Interaction of the F-15 Airplane. NASA, Technical Paper 2374, California, USA (1984)

Rotating Detonation Engine Injection Velocity Limit and Nozzle Effects on Its Propulsion Performance Jian-Ping Wang and Ye-Tao Shao

Abstract Three-dimensional numerical simulation of a rotating detonation engine (RDE) is carried out in coaxial tube chamber to reveal its physical characteristics. Multi-cycle process of RDE is numerically obtained, and it qualitatively agrees with former experimental results. Some key problems about RDE achievements such as fuel injection, pre-combustion, detonation structure are discussed. At last, we made a propulsion performance analysis about RDE by several different numerical cases.

1 Introduction The concept of a RDE, which is also called a continuous detonation wave engine (CDWE), was first achieved experimentally in short time by Voitsekhoviskii [7] in the early 1960s. In recent years, RDEs have been extensively studied from an experimental viewpoint by Bykovskii et al. [1–3]. The serial experiments achieved both liquid and gas fuel detonation in combustors with different shapes and with supersonic or subsonic injection flow. From a numerical viewpoint, some twodimensional simulations [4, 5, 9] and three-dimensional simulations [6, 8] have been done for the flow field of RDEs as well as different aspects of their propulsive performance. All of the above experimental and numerical investigations show the advantages and applicability of the RDE concept. An RDE can work continuously under various operation conditions of subsonic and supersonic injection. However, the limitation of injection velocity beyond which RDEs no longer work and the accompanying changes in the DW propagation mode have not yet been investigated. In the present study, we investigate these questions by computing a series of cases with injection velocities ranging from 50 to 2,000 m s−1 .

J.-P. Wang (B) State Key Laboratory of Turbulence and Complex Systems, Department of Mechanics and Aerospace Engineering, Peking University, Beijing, China 100871 e-mail: [email protected]


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Furthermore, a detail propulsion performance analysis for an RDE with different type of nozzles is carried out. RDEs with four types of nozzle, namely constant area nozzle, Laval nozzle, diverging nozzle and converging nozzle are numerically simulated to investigate their propulsion performance.

2 Physical Model and Numerical Method A one-step chemical kinetic model was used in this simulation. Three-dimensional Euler equations in generalized coordinates are used as governing equations. The geometry of the chamber and the overview of the continuous propagation processes of an RDE with an injection velocity of 500 m s−1 are illustrated in Fig. 1. The inner radius of the chamber was 40 mm, the outer radius was 53 mm, and the tube length was 60 mm. The head wall was closed but was perforated with small, uniformly distributed ports or slits, which were used to inject combustible gas into the chamber. When the head-wall pressure pw exceeded the injection stagnation pressure, the reaction mixture could not be injected into the chamber and a rigid wall condition was set locally. Otherwise, we assumed that the injection flow maintained a constant state of 0.103 MPa, 300 K with the injection velocity from 50 to 2,000 m s−1 for the different cases.

Fig. 1 Continuous propagation processes an RDW from 0 to 850 μs. Panels (a) and (b) show the pressure contour, and panels (c) and (d) show the contour of the reaction progress parameter β. Point 1 indicates CJ detonation, point 2 is the DW, point 3 is the oblique shock wave, point 4 is the detonation product, point 5 is the new injected fresh gas mixture, and point 6 is the deflagration interface front

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3 Results and Discussions 3.1 Injection Velocity Limit Figure 1a, b show the pressure contour at t = 0 and 100 μs, respectively. The DW is ignited by a branching, pre-detonation tube that is connected tangentially to the outer wall of the combustor. In numerical simulations, a section of classical CJ detonation is set, as shown in Fig. 1a, then is substituted of the pre-detonation ignition for simplicity. After ignition, the DW propagates azimuthally around the combustor. Because the DW’s velocity is approximately 2,000 m s−1 , the injection fuel velocity is 500 m s−1 . However, the velocity of the deflagration wave at the interface between the detonation product and the new injected fuel is approximately several meters per second. Only a thin layer of new injected fuel is burnt out, so that enough injected fuel could maintain the unreacted state to support a continuously propagating DW. Figure 1c, d show the reaction progress parameter contour β at 150 and 850 μs, respectively. These figures show that, at 850 μs, the DW has propagated more than six rounds, and it can continuously propagate for a long time. The numerical results in Fig. 1 agree well qualitatively with existing experimental results [3]. The DW maintains an acute angle of approximately 20 with respect to the fuel injection direction so that it moves against the injection flow direction, thereby avoiding being blown downstream. Because of this inclination, the azimuthal branch velocity is lower than the classical CJ velocity. Figure 2 shows the history of the pressure and

Pressure /MPa

4 3 2 1 0

0

200

4000

Temperature /K

400

600

800

600

800

Time /μs

3000 2000 1000 0

0

200

400

Time /μs

Fig. 2 A single history of the evolution of point pressure and temperature at the headpiece (x = 45 mm, y = 0 mm, z = 10 mm) of the combustor from 0 to 850 μs

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z/m

0.06

(a) Win = 100m/s

0.04

θ

Computational Fluid Dynamics 2010

Computational Fluid Dynamics

Computational fluid dynamics